import{_ as a,c as x,a as l,b as m,d as s,o as j}from"./app-Bj0iHonY.js";const n="/assets/math_relatively_prime-DfVfRDyq.png",e="/assets/math_euclid-BsqmJaLP.png",t={};function i(o,c){return j(),x("div",null,c[0]||(c[0]=[l('<h1 id="数论-一" tabindex="-1"><a class="header-anchor" href="#数论-一"><span>数论（一）</span></a></h1><hr><h2 id="_1-同余系统" tabindex="-1"><a class="header-anchor" href="#_1-同余系统"><span>1.同余系统</span></a></h2><h3 id="_1-1-最大公约数-greatest-common-divisor" tabindex="-1"><a class="header-anchor" href="#_1-1-最大公约数-greatest-common-divisor"><span>1.1 最大公约数(<a href="https://en.wikipedia.org/wiki/Greatest_common_divisor" target="_blank" rel="noopener noreferrer">Greatest common divisor</a>)</span></a></h3>',4),m("h4",{id:"_1-1-1-表示x和y的最大公约数-也叫最大公因数-例如",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-1-1-表示x和y的最大公约数-也叫最大公因数-例如"},[m("span",null,[s("1.1.1 "),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y"),m("mo",{stretchy:"false"},")")])])]),s("表示x和y的最大公约数，也叫最大公因数，例如"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c36"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"6"),m("mo",null,","),m("mn",null,"15"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"3")])])])])])],-1),m("p",null,"欧几里得对“公约”的解释，给定两条长度不同的线段 a 和 b ，如果能够找到第三条线段 c ，它既可以度量 a ，又可以度量 b ，我们就说 a 和 b 是可公约的(commensurable)，最大可能的c就是a和b的最大公约数",-1),m("h4",{id:"_1-1-2-如果两个数的最大公约数是1-那么这两个数成是互质的-relatively-prime-比如说",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-1-2-如果两个数的最大公约数是1-那么这两个数成是互质的-relatively-prime-比如说"},[m("span",null,[s("1.1.2 如果两个数的最大公约数是1，那么这两个数成是互质的(relatively prime)，比如说"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"7"),m("mo",null,","),m("mn",null,"10"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])])])])],-1),m("p",null,"互质的几何意义，如果 a 和 b 互质，这就意味着分数 a / b 已经不能再约分了，意味着 a × b 的棋盘的对角线不会经过中间的任何交叉点，意味着循环长度分别为 a 和 b 的两个周期性事件一同上演，则新的循环长度最短为 a × b 。",-1),m("figure",null,[m("img",{src:n,alt:"",tabindex:"0",loading:"lazy"}),m("figcaption")],-1),m("h4",{id:"_1-1-3-最大公约数一般使用欧几里得碾压相除法-euclidean-algorithm-计算",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-1-3-最大公约数一般使用欧几里得碾压相除法-euclidean-algorithm-计算"},[m("span",null,[s("1.1.3 最大公约数一般使用欧几里得碾压相除法("),m("a",{href:"https://en.wikipedia.org/wiki/Euclidean_algorithm",target:"_blank",rel:"noopener noreferrer"},"Euclidean algorithm"),s(")计算")])])],-1),m("p",null,"欧几里得算法使用了如下递归式",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",null,","),m("mi",null,"m"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"m"),m("mo",null,","),m("mi",null,"n"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"m"),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c36"}),m("mjx-c",{class:"mjx-c39"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"69"),m("mo",null,","),m("mn",null,"21"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"3")])])]),s("的计算方法"),m("br"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c36"}),m("mjx-c",{class:"mjx-c39"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c36"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c36"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"69"),m("mo",null,","),m("mn",null,"21"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"21"),m("mo",null,","),m("mn",null,"6"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mn",null,"6"),m("mo",null,","),m("mn",null,"3"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"3")])])])],-1),m("figure",null,[m("img",{src:e,alt:"",tabindex:"0",loading:"lazy"}),m("figcaption")],-1),m("h3",{id:"_1-2-最小公倍数-least-common-multiple",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-2-最小公倍数-least-common-multiple"},[m("span",null,[s("1.2 最小公倍数("),m("a",{href:"https://en.wikipedia.org/wiki/Least_common_multiple",target:"_blank",rel:"noopener noreferrer"},"Least Common Multiple"),s(")")])])],-1),m("p",null,[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D459 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"l"),m("mi",null,"c"),m("mi",null,"m"),m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y"),m("mo",{stretchy:"false"},")")])])]),s("表示x和y的最小公倍数,例如"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D459 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c39"}),m("mjx-c",{class:"mjx-c30"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"l"),m("mi",null,"c"),m("mi",null,"m"),m("mo",{stretchy:"false"},"("),m("mn",null,"45"),m("mo",null,","),m("mn",null,"30"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"90")])])])],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D459 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mi",null,"l"),m("mi",null,"c"),m("mi",null,"m"),m("mo",{stretchy:"false"},"("),m("mi",null,"x"),m("mo",null,","),m("mi",null,"y"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mi",null,"x"),m("mi",null,"y")])])],-1),m("h3",{id:"_1-3-模运算",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-3-模运算"},[m("span",null,"1.3 模运算")])],-1),m("h4",{id:"_1-3-1-是除以的余数-例如",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-3-1-是除以的余数-例如"},[m("span",null,[s("1.3.1 "),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"n")])])])]),s(" 是"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("除以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的余数,例如"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"13"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"5")]),m("mo",null,"="),m("mn",null,"3")])])])])])],-1),m("h4",{id:"_1-3-2-模运算符合交换律-结合律和分配律",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-3-2-模运算符合交换律-结合律和分配律"},[m("span",null,"1.3.2 模运算符合交换律，结合律和分配律")])],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{style:{"min-width":"28.381em"}},[m("mjx-table",null,[m("mjx-itable",null,[m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-bottom":"0.15em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-tstrut")])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[m("mtr",null,[m("mtd",null,[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"+"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"+"),m("mo",{stretchy:"false"},"("),m("mi",null,"b"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])]),m("mtr",null,[m("mtd",null,[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"−"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mo",{stretchy:"false"},"("),m("mi",null,"b"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])]),m("mtr",null,[m("mtd",null,[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"×"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mo",{stretchy:"false"},"("),m("mi",null,"b"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])]),m("mtr",null,[m("mtd",null,[m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"×"),m("mo",{stretchy:"false"},"("),m("mi",null,"b"),m("mo",null,"+"),m("mi",null,"c"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"×"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"+"),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,"×"),m("mi",null,"c"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])])])])],-1),m("h4",{id:"_1-3-3-如何计算",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-3-3-如何计算"},[m("span",null,[s("1.3.3 如何计算"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mi",null,"x")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])])])])],-1),m("p",null,[s("如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("是2的幂次方，例如16，那么 "),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c36"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"16")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])]),m("br"),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("不是2的幂次方，例如"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c35"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"25")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])]),m("br"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c36"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c38"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"25"),m("mo",null,"="),m("mn",null,"16"),m("mo",null,"+"),m("mn",null,"8"),m("mo",null,"+"),m("mn",null,"1"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mn",null,"2"),m("mo",null,"+"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"+"),m("mn",null,"1")])])]),m("br"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c35"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"25")])]),m("mo",null,"="),m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",{stretchy:"false"},"("),m("mn",null,"2"),m("mo",null,"+"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"+"),m("mn",null,"1")])]),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",null,"×"),m("mi",null,"a"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",null,"×"),m("mi",null,"a")])])]),m("br"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c35"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"25")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mi",null,"a"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mn",null,"2")]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"×"),m("mi",null,"a"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"n")])])])],-1),m("h3",{id:"_1-4-同余等式",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_1-4-同余等式"},[m("span",null,"1.4 同余等式")])],-1),m("p",null,[s("如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"b")])])]),s("除"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的余数相同，那么记为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mo",null,"≡"),m("mi",null,"b"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mo",null,"≡"),m("mi",null,"b"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("，那么"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"b"),m("mo",null,"≡"),m("mi",null,"a"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D450 TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mo",null,"≡"),m("mi",null,"b"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,","),m("mi",null,"b"),m("mo",null,"≡"),m("mi",null,"c"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("，那么记为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mo",null,"≡"),m("mi",null,"b"),m("mo",null,"≡"),m("mi",null,"a"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("mo",null,"≡"),m("msub",null,[m("mi",null,"b"),m("mn",null,"1")]),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"m"),m("mo",{stretchy:"false"},")"),m("mo",null,","),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",null,"≡"),m("msub",null,[m("mi",null,"b"),m("mn",null,"2")]),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"m"),m("mo",{stretchy:"false"},")"),m("mo",null,","),m("mo",null,"…"),m("mo",null,","),m("msub",null,[m("mi",null,"a"),m("mi",null,"n")]),m("mo",null,"≡"),m("msub",null,[m("mi",null,"b"),m("mi",null,"n")]),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"m"),m("mo",{stretchy:"false"},")")])])]),s("，那么")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{style:{"min-width":"22.47em"}},[m("mjx-table",null,[m("mjx-itable",null,[m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-bottom":"0.15em"}},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em"}},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-tstrut")])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[m("mtr",null,[m("mtd",null,[m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("mo",null,"+"),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",null,"+"),m("mo",null,"…"),m("mo",null,"+"),m("msub",null,[m("mi",null,"a"),m("mi",null,"n")])]),m("mtd",null,[m("mi"),m("mo",null,"≡"),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"b"),m("mn",null,"1")]),m("mo",null,"+"),m("msub",null,[m("mi",null,"b"),m("mn",null,"2")]),m("mo",null,"+"),m("mo",null,"…"),m("mo",null,"+"),m("msub",null,[m("mi",null,"b"),m("mi",null,"n")]),m("mo",{stretchy:"false"},")"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"m"),m("mo",{stretchy:"false"},")")])]),m("mtr",null,[m("mtd",null,[m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",null,"…"),m("msub",null,[m("mi",null,"a"),m("mi",null,"n")])]),m("mtd",null,[m("mi"),m("mo",null,"≡"),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"b"),m("mn",null,"1")]),m("msub",null,[m("mi",null,"b"),m("mn",null,"2")]),m("mo",null,"…"),m("msub",null,[m("mi",null,"b"),m("mi",null,"n")]),m("mo",{stretchy:"false"},")"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"m"),m("mo",{stretchy:"false"},")")])])])])])],-1),l('<h2 id="_2-中国剩余定理" tabindex="-1"><a class="header-anchor" href="#_2-中国剩余定理"><span>2.中国剩余定理</span></a></h2><h3 id="_2-1-孙子算经中的问题" tabindex="-1"><a class="header-anchor" href="#_2-1-孙子算经中的问题"><span>2.1 孙子算经中的问题</span></a></h3><blockquote><p>“今有物，不知其数。三、三数之，剩二；五、五数之，剩三；七、七数之，剩二。问物几何？答曰：二十三。”<br> 《孙子算经》卷下第二十六问</p></blockquote><h3 id="_2-2-数学解释" tabindex="-1"><a class="header-anchor" href="#_2-2-数学解释"><span>2.2 数学解释</span></a></h3>',4),m("p",null,[s("有"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"m")])])]),s("个两两互质的数，其乘积为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D443 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"P")])])]),s("，有一个未知数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"M")])])]),s("，如果我们已知"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"M")])])]),s("分别除以这"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"m")])])]),s("个数所得的余数，那么在0到"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D443 TEX-I"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2013"})])]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"P"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"–")]),m("mn",null,"1")])])]),s("的范围内，我们可以唯一地确定这个"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"M")])])])],-1),m("h3",{id:"_2-3-直觉",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_2-3-直觉"},[m("span",null,"2.3 直觉")])],-1),m("p",null,"两个除数的情况为例，假设两个除数分别是 4 和 7。下表显示的就是各自然数除以 4 和除以 7 的余数情况",-1),m("table",null,[m("thead",null,[m("tr",null,[m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c30"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"0")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"1")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"2")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"3")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"4")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"5")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c36"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"6")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"7")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c38"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"8")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c39"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"9")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"10")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"11")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c32"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"12")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"13")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"14")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c35"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"15")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c36"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"16")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"17")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c38"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"18")])])])]),m("th",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c39"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"19")])])])])])]),m("tbody",null,[m("tr",null,[m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"4")])])])]),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3")]),m("tr",null,[m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"7")])])])]),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4"),m("td",{style:{"text-align":"center"}},"5"),m("td",{style:{"text-align":"center"}},"6"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4"),m("td",{style:{"text-align":"center"}},"5"),m("td",{style:{"text-align":"center"}},"6"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4"),m("td",{style:{"text-align":"center"}},"5")]),m("tr",null,[m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c30"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"20")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"21")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c32"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"22")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"23")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"24")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c35"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"25")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c36"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"26")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"27")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c38"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"28")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c39"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"29")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c30"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"30")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"31")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c32"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"32")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c33"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"33")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"34")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c35"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"35")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c36"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"36")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"37")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c38"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"38")])])])]),m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c39"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"39")])])])])]),m("tr",null,[m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"4")])])])]),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3")]),m("tr",null,[m("td",{style:{"text-align":"center"}},[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"7")])])])]),m("td",{style:{"text-align":"center"}},"6"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4"),m("td",{style:{"text-align":"center"}},"5"),m("td",{style:{"text-align":"center"}},"6"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4"),m("td",{style:{"text-align":"center"}},"5"),m("td",{style:{"text-align":"center"}},"6"),m("td",{style:{"text-align":"center"}},"0"),m("td",{style:{"text-align":"center"}},"1"),m("td",{style:{"text-align":"center"}},"2"),m("td",{style:{"text-align":"center"}},"3"),m("td",{style:{"text-align":"center"}},"4")])])],-1),m("p",null,[s("可以看出，"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"4"),m("mo",null,","),m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"7")])])]),s("的结果以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c38"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"4"),m("mi",null,"×"),m("mn",null,"7"),m("mo",null,"="),m("mn",null,"28")])])]),s("长度循环，每种特性的组合都在这28个结果中出现，所以一旦确定"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"4")])])]),s("和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"7")])])]),s("的余数，就一定能唯一判定出"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"i")])])])],-1),m("h3",{id:"_2-4-数学表达",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_2-4-数学表达"},[m("span",null,"2.4 数学表达")])],-1),m("p",null,[s("已知"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msub",null,[m("mi",null,"m"),m("mn",null,"1")]),m("mo",null,","),m("msub",null,[m("mi",null,"m"),m("mn",null,"2")]),m("mo",null,","),m("mo",null,"…"),m("mo",null,","),m("msub",null,[m("mi",null,"m"),m("mi",null,"k")])])])]),s("是两两互质的正整数，有未知数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("满足同余方程组")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace"),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace"),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c22EF"})]),m("mjx-mspace"),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace")]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"x"),m("mo",null,"≡"),m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("msub",null,[m("mi",null,"m"),m("mn",null,"1")]),m("mo",{stretchy:"false"},")"),m("mspace",{linebreak:"newline"}),m("mi",null,"x"),m("mo",null,"≡"),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("msub",null,[m("mi",null,"m"),m("mn",null,"2")]),m("mo",{stretchy:"false"},")"),m("mspace",{linebreak:"newline"}),m("mo",null,"⋯"),m("mspace",{linebreak:"newline"}),m("mi",null,"x"),m("mo",null,"≡"),m("msub",null,[m("mi",null,"a"),m("mi",null,"k")]),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("msub",null,[m("mi",null,"m"),m("mi",null,"k")]),m("mo",{stretchy:"false"},")"),m("mspace",{linebreak:"newline"})])])],-1),m("p",null,"那么",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.316em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"x"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("msub",null,[m("mi",null,"M"),m("mn",null,"1")]),m("msubsup",null,[m("mi",null,"M"),m("mn",null,"1"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"+"),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("msub",null,[m("mi",null,"M"),m("mn",null,"2")]),m("msubsup",null,[m("mi",null,"M"),m("mn",null,"2"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"+"),m("mo",null,"…"),m("mo",null,"+"),m("msub",null,[m("mi",null,"a"),m("mi",null,"k")]),m("msub",null,[m("mi",null,"M"),m("mi",null,"k")]),m("msubsup",null,[m("mi",null,"M"),m("mi",null,"k"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])]),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"M")])])])],-1),m("p",null,[s("其中"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msub",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c22C5"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c22C5"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c22C5"})]),m("mjx-msub",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2F"})])]),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"M"),m("mo",null,"="),m("msub",null,[m("mi",null,"m"),m("mn",null,"1")]),m("mo",null,"⋅"),m("msub",null,[m("mi",null,"m"),m("mn",null,"2")]),m("mo",null,"⋅"),m("mo",null,"…"),m("mo",null,"⋅"),m("msub",null,[m("mi",null,"m"),m("mi",null,"k")]),m("mo",null,","),m("msub",null,[m("mi",null,"M"),m("mi",null,"i")]),m("mo",null,"="),m("mi",null,"M"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"/")]),m("msub",null,[m("mi",null,"m"),m("mi",null,"i")])])])]),s(", 而"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.294em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msubsup",null,[m("mi",null,"M"),m("mi",null,"i"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])])])])]),s("是"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msub",null,[m("mi",null,"M"),m("mi",null,"i")])])])]),s("的逆元（见3.3节），也就是"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])]),m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.294em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D456 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msub",null,[m("mi",null,"M"),m("mi",null,"i")]),m("msubsup",null,[m("mi",null,"M"),m("mi",null,"i"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("msub",null,[m("mi",null,"m"),m("mi",null,"i")]),m("mo",{stretchy:"false"},")")])])])],-1),m("h3",{id:"_2-5-举例-孙子算经中的问题",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_2-5-举例-孙子算经中的问题"},[m("span",null,"2.5 举例，孙子算经中的问题")])],-1),m("p",null,[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45A TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msub",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msub",null,[m("mi",null,"m"),m("mn",null,"1")]),m("mo",null,"="),m("mn",null,"3"),m("mo",null,","),m("msub",null,[m("mi",null,"m"),m("mn",null,"2")]),m("mo",null,"="),m("mn",null,"5"),m("mo",null,","),m("msub",null,[m("mi",null,"m"),m("mn",null,"3")]),m("mo",null,"="),m("mn",null,"7"),m("mo",null,","),m("msub",null,[m("mi",null,"a"),m("mn",null,"1")]),m("mo",null,"="),m("mn",null,"2"),m("mo",null,","),m("msub",null,[m("mi",null,"a"),m("mn",null,"2")]),m("mo",null,"="),m("mn",null,"3"),m("mo",null,","),m("msub",null,[m("mi",null,"a"),m("mn",null,"3")]),m("mo",null,"="),m("mn",null,"2")])])]),s("，那么")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mtable",{style:{"min-width":"26.261em"}},[m("mjx-table",null,[m("mjx-itable",null,[m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em","margin-left":"-0.081em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D440 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.295em","margin-left":"-0.081em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD",style:{"margin-left":"0.191em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em","padding-bottom":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-TeXAtom",{texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-tstrut")])]),m("mjx-mtr",null,[m("mjx-mtd",{style:{"text-align":"right","padding-right":"0","padding-top":"0.15em"}},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-tstrut")]),m("mjx-mtd",{style:{"text-align":"left","padding-left":"0","padding-top":"0.15em"}},[m("mjx-mi",{class:"mjx-n"}),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c32"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-tstrut")])])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mtable",{displaystyle:"true",columnalign:"right left",columnspacing:"0em",rowspacing:"3pt"},[m("mtr",null,[m("mtd",null,[m("mi",null,"M")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mn",null,"3"),m("mo",null,"×"),m("mn",null,"5"),m("mo",null,"×"),m("mn",null,"7"),m("mo",null,"="),m("mn",null,"105")])]),m("mtr",null,[m("mtd",null,[m("msub",null,[m("mi",null,"M"),m("mn",null,"1")])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mn",null,"5"),m("mo",null,"×"),m("mn",null,"7"),m("mo",null,"="),m("mn",null,"35")])]),m("mtr",null,[m("mtd",null,[m("msub",null,[m("mi",null,"M"),m("mn",null,"2")])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mn",null,"3"),m("mo",null,"×"),m("mn",null,"7"),m("mo",null,"="),m("mn",null,"21")])]),m("mtr",null,[m("mtd",null,[m("msub",null,[m("mi",null,"M"),m("mn",null,"3")])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mn",null,"3"),m("mo",null,"×"),m("mn",null,"5"),m("mo",null,"="),m("mn",null,"15")])]),m("mtr",null,[m("mtd",null,[m("msubsup",null,[m("mi",null,"M"),m("mn",null,"1"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("msup",null,[m("mn",null,"35"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"3"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"2")]),m("mo",null,"="),m("mn",null,"2")])]),m("mtr",null,[m("mtd",null,[m("msubsup",null,[m("mi",null,"M"),m("mn",null,"2"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("msup",null,[m("mn",null,"21"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"5"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"5")]),m("mo",null,"="),m("mn",null,"1")])]),m("mtr",null,[m("mtd",null,[m("msubsup",null,[m("mi",null,"M"),m("mn",null,"3"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mo",null,"−"),m("mn",null,"1")])])]),m("mtd",null,[m("mi"),m("mo",null,"="),m("msup",null,[m("mn",null,"15"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"7"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"7")]),m("mo",null,"="),m("mn",null,"1")])]),m("mtr",null,[m("mtd",null,[m("mi",null,"x")]),m("mtd",null,[m("mi"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"35"),m("mo",null,"×"),m("mn",null,"2"),m("mo",null,"+"),m("mn",null,"3"),m("mo",null,"×"),m("mn",null,"21"),m("mo",null,"×"),m("mn",null,"1"),m("mo",null,"+"),m("mn",null,"2"),m("mo",null,"×"),m("mn",null,"15"),m("mo",null,"×"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mn",null,"105"),m("mo",null,"="),m("mn",null,"23")])])])])])],-1),m("h2",{id:"_3-欧拉定理",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-欧拉定理"},[m("span",null,"3.欧拉定理")])],-1),m("h3",{id:"_3-1-欧拉函数",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-1-欧拉函数"},[m("span",null,"3.1 欧拉函数")])],-1),m("p",null,[s("任意给定正整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("，小于等于"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的正整数之中，与n构成互质关系的数的个数记为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])])],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1")])])],-1),m("p",null,[s("如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p")])])]),s("是质数，那么"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"p"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mi",null,"p"),m("mo",null,"−"),m("mn",null,"1")])])]),s("， "),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-msup",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"p"),m("mi",null,"k")]),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("msup",null,[m("mi",null,"p"),m("mi",null,"k")]),m("mo",null,"−"),m("msup",null,[m("mi",null,"p"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"k"),m("mo",null,"−"),m("mn",null,"1")])])])])]),m("br"),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n"),m("mo",null,"="),m("mi",null,"p"),m("mi",null,"q"),m("mo",null,","),m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"p"),m("mo",{stretchy:"false"},")"),m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"q"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",{stretchy:"false"},"("),m("mi",null,"p"),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},"("),m("mi",null,"q"),m("mo",null,"−"),m("mn",null,"1"),m("mo",{stretchy:"false"},")")])])]),m("br"),s(" 如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msubsup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"0"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])]),m("mjx-msubsup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.296em","margin-left":"0"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])]),m("mjx-spacer",{style:{"margin-top":"0.18em"}}),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-msubsup",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.247em","margin-left":"0"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45F TEX-I"})])]),m("mjx-spacer",{style:{"margin-top":"0.29em"}}),m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45F TEX-I"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n"),m("mo",null,"="),m("msubsup",null,[m("mi",null,"p"),m("mn",null,"1"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"k"),m("mn",null,"1")])]),m("msubsup",null,[m("mi",null,"p"),m("mn",null,"2"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"k"),m("mn",null,"2")])]),m("mo",null,"…"),m("msubsup",null,[m("mi",null,"p"),m("mi",null,"r"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"k"),m("mi",null,"r")])])])])]),s("，那么")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mfrac",{space:"3"},[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])])])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mfrac",{space:"3"},[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mfrac",{space:"3"},[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-msub",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-script",{style:{"vertical-align":"-0.15em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45F TEX-I"})])])])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mi",null,"n"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",null,"−"),m("mfrac",null,[m("mn",null,"1"),m("msub",null,[m("mi",null,"p"),m("mn",null,"1")])]),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",null,"−"),m("mfrac",null,[m("mn",null,"1"),m("msub",null,[m("mi",null,"p"),m("mn",null,"2")])]),m("mo",{stretchy:"false"},")"),m("mo",null,"…"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",null,"−"),m("mfrac",null,[m("mn",null,"1"),m("msub",null,[m("mi",null,"p"),m("mi",null,"r")])]),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,[s("比如"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-msup",{space:"3"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c32"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c33"}),m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mfrac",{space:"3"},[m("mjx-frac",null,[m("mjx-num",null,[m("mjx-nstrut"),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line"),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut"),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c33"})])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mfrac",{space:"3"},[m("mjx-frac",null,[m("mjx-num",null,[m("mjx-nstrut"),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c31"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line"),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut"),m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c37"})])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c37"}),m("mjx-c",{class:"mjx-c35"}),m("mjx-c",{class:"mjx-c36"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"1323"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mn",null,"3"),m("mn",null,"3")]),m("mo",null,"×"),m("msup",null,[m("mn",null,"7"),m("mn",null,"2")]),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"1323"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",null,"−"),m("mfrac",null,[m("mn",null,"1"),m("mn",null,"3")]),m("mo",{stretchy:"false"},")"),m("mo",{stretchy:"false"},"("),m("mn",null,"1"),m("mo",null,"−"),m("mfrac",null,[m("mn",null,"1"),m("mn",null,"7")]),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"756")])])])],-1),m("h3",{id:"_3-2-欧拉定理",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-2-欧拉定理"},[m("span",null,"3.2 欧拉定理")])],-1),m("p",null,[s("如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("互质，那么")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])]),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])],-1),m("h4",{id:"_3-2-1",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-2-1"},[m("span",null,"3.2.1")])],-1),m("p",null,[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c34"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"10"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mn",null,"4")])])]),s("，"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c34"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c34"}),m("mjx-c",{class:"mjx-c30"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"10"),m("mo",{stretchy:"false"},")")])]),m("mo",null,"≡"),m("msup",null,[m("mn",null,"7"),m("mn",null,"4")]),m("mo",null,"≡"),m("mn",null,"2401"),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"10"),m("mo",{stretchy:"false"},")")])])])],-1),m("h4",{id:"_3-2-2",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-2-2"},[m("span",null,"3.2.2")])],-1),m("p",null,[m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c34"})])])]),m("mjx-msup",null,[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-munder",{space:"4"},[m("mjx-row",null,[m("mjx-base",null,[m("mjx-TeXAtom",{texclass:"OP"},[m("mjx-munder",null,[m("mjx-row",null,[m("mjx-base",null,[m("mjx-mrow",null,[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c34"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-msup",{space:"3"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c34"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-msup",{space:"3"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c34"})])])])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.105em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-stretchy-h",{style:{width:"7.649em"},class:"mjx-c23DF"},[m("mjx-beg",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-mid",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-end",null,[m("mjx-c")])])])])])])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.062em","padding-left":"3.638em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mtext",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6B"})])])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-munder",{space:"4"},[m("mjx-row",null,[m("mjx-base",null,[m("mjx-TeXAtom",{texclass:"OP"},[m("mjx-munder",null,[m("mjx-row",null,[m("mjx-base",null,[m("mjx-mrow",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.105em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-stretchy-h",{style:{width:"6.339em"},class:"mjx-c23DF"},[m("mjx-beg",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-mid",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-end",null,[m("mjx-c")])])])])])])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.062em","padding-left":"2.983em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mtext",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6B"})])])])])]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"4"),m("mi",null,"k")])]),m("mo",null,"≡"),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mn",null,"7"),m("mn",null,"4")]),m("msup",null,[m("mo",{stretchy:"false"},")"),m("mi",null,"k")]),m("mo",null,"≡"),m("munder",null,[m("mrow",{"data-mjx-texclass":"OP"},[m("munder",null,[m("mrow",null,[m("msup",null,[m("mn",null,"7"),m("mn",null,"4")]),m("mo",null,"×"),m("msup",null,[m("mn",null,"7"),m("mn",null,"4")]),m("mo",null,"×"),m("mo",null,"…"),m("mo",null,"×"),m("msup",null,[m("mn",null,"7"),m("mn",null,"4")])]),m("mo",null,"⏟")])]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mtext",null,"k")])]),m("mo",null,"≡"),m("munder",null,[m("mrow",{"data-mjx-texclass":"OP"},[m("munder",null,[m("mrow",null,[m("mn",null,"1"),m("mo",null,"×"),m("mn",null,"1"),m("mo",null,"×"),m("mo",null,"…"),m("mo",null,"×"),m("mn",null,"1")]),m("mo",null,"⏟")])]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mtext",null,"k")])]),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("，所以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"4"),m("mi",null,"k")])])])])]),s("尾数一定是1")],-1),m("h4",{id:"_3-2-3",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-2-3"},[m("span",null,"3.2.3")])],-1),m("p",null,[s("由于"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msup",{space:"2"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c39"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"4"),m("mi",null,"k")])]),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"10"),m("mo",{stretchy:"false"},")"),m("mo",null,","),m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"2")])]),m("mo",null,"≡"),m("mn",null,"9"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"10"),m("mo",{stretchy:"false"},")")])])]),s(", 所以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"}),m("mjx-c",{class:"mjx-c35"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-munder",null,[m("mjx-row",null,[m("mjx-base",null,[m("mjx-TeXAtom",{texclass:"OP"},[m("mjx-munder",null,[m("mjx-row",null,[m("mjx-base",null,[m("mjx-mrow",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2026"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.105em"}},[m("mjx-mo",{class:"mjx-n"},[m("mjx-stretchy-h",{style:{width:"6.339em"},class:"mjx-c23DF"},[m("mjx-beg",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-mid",null,[m("mjx-c")]),m("mjx-ext",null,[m("mjx-c")]),m("mjx-end",null,[m("mjx-c")])])])])])])])])]),m("mjx-row",null,[m("mjx-under",{style:{"padding-top":"0.062em","padding-left":"2.816em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c35"}),m("mjx-c",{class:"mjx-c35"})])])])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c39"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c30"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"222")])]),m("mo",null,"≡"),m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"4"),m("mo",null,"×"),m("mn",null,"55"),m("mo",null,"+"),m("mn",null,"2")])]),m("mo",null,"≡"),m("mo",{stretchy:"false"},"("),m("munder",null,[m("mrow",{"data-mjx-texclass":"OP"},[m("munder",null,[m("mrow",null,[m("mn",null,"1"),m("mo",null,"×"),m("mn",null,"1"),m("mo",null,"×"),m("mo",null,"…"),m("mo",null,"×"),m("mn",null,"1")]),m("mo",null,"⏟")])]),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"55")])]),m("mo",null,"×"),m("mn",null,"9"),m("mo",{stretchy:"false"},")"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"10"),m("mo",{stretchy:"false"},")")])])]),s("，所以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c37"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c32"}),m("mjx-c",{class:"mjx-c32"})])])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"7"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mn",null,"222")])])])])]),s("尾数为9")],-1),m("h3",{id:"_3-3-模反元素-逆元",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-3-模反元素-逆元"},[m("span",null,"3.3 模反元素（逆元）")])],-1),m("p",null,[s("如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("和"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("互质，那么"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mi",null,"x"),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("有解，"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"x")])])]),s("称为"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("模"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"n")])])]),s("的模反元素")],-1),m("h4",{id:"_3-3-1",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-3-1"},[m("span",null,"3.3.1")])],-1),m("p",null,[s("用欧拉定理证明"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-cD7"})]),m("mjx-msup",{space:"3"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])]),m("mo",null,"≡"),m("mi",null,"a"),m("mo",null,"×"),m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"n"),m("mo",{stretchy:"false"},")")])])]),s("，所以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45B TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mi",null,"n"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"+"),m("mi",null,"k"),m("mi",null,"n")])])]),s("都是a的模反元素")],-1),m("h4",{id:"_3-3-2",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-3-2"},[m("span",null,"3.3.2")])],-1),m("p",null,[s("例如求解同余方程"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"3"),m("mi",null,"x"),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"11"),m("mo",{stretchy:"false"},")")])])]),s("，那么"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D719 TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-msup",{space:"4"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c33"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mn",{class:"mjx-n",size:"s"},[m("mjx-c",{class:"mjx-c39"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c39"}),m("mjx-c",{class:"mjx-c36"}),m("mjx-c",{class:"mjx-c38"}),m("mjx-c",{class:"mjx-c33"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mn",{class:"mjx-n",space:"2"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mn",null,"3"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"ϕ"),m("mo",{stretchy:"false"},"("),m("mn",null,"11"),m("mo",{stretchy:"false"},")"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"≡"),m("msup",null,[m("mn",null,"3"),m("mn",null,"9")]),m("mo",null,"≡"),m("mn",null,"19683"),m("mo",null,"≡"),m("mn",null,"4"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mn",null,"11"),m("mo",{stretchy:"false"},")")])])]),s("，所以"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c34"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mn",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c31"}),m("mjx-c",{class:"mjx-c31"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D458 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mn",null,"4"),m("mo",null,"+"),m("mn",null,"11"),m("mi",null,"k")])])]),s("都是同余方程的解")],-1),m("h3",{id:"_3-4-费马小定理",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_3-4-费马小定理"},[m("span",null,"3.4 费马小定理")])],-1),m("p",null,[s("根据欧拉定理，当"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p")])])]),s("是一个质数时，对于任意正整数"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a")])])]),s("，有"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mi",{class:"mjx-i",size:"s"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mspace",{style:{width:"0.444em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("msup",null,[m("mi",null,"a"),m("mi",null,"p")]),m("mo",null,"≡"),m("mi",null,"a"),m("mspace",{width:"0.444em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"p"),m("mo",{stretchy:"false"},")")])])]),s("，如果"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2224 TEX-A"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"p"),m("mo",null,"∤"),m("mi",null,"a")])])]),s("，那么")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-TeXAtom",{size:"s",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mn",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c31"})])])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2261"})]),m("mjx-mn",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c31"})]),m("mjx-mspace",{style:{width:"1em"}}),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mspace",{style:{width:"0.333em"}}),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D45D TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("msup",null,[m("mi",null,"a"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"p"),m("mo",null,"−"),m("mn",null,"1")])]),m("mo",null,"≡"),m("mn",null,"1"),m("mspace",{width:"1em"}),m("mo",{stretchy:"false"},"("),m("mi",null,"mod"),m("mspace",{width:"0.333em"}),m("mi",null,"p"),m("mo",{stretchy:"false"},")")])])],-1),m("h2",{id:"_4-扩展碾压相除法",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_4-扩展碾压相除法"},[m("span",null,"4.扩展碾压相除法")])],-1),m("h3",{id:"_4-1-裴蜀定理",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_4-1-裴蜀定理"},[m("span",null,"4.1 裴蜀定理")])],-1),m("p",null,[s("对于方程"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mi",null,"x"),m("mo",null,"+"),m("mi",null,"b"),m("mi",null,"y"),m("mo",null,"="),m("mi",null,"d")])])]),s("，当且仅当"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c2223"})]),m("mjx-TeXAtom",{space:"4",texclass:"ORD"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D451 TEX-I"})])])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,","),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",null,"∣"),m("mrow",{"data-mjx-texclass":"ORD"},[m("mi",null,"d")])])])]),s("时，方程有整数解")],-1),m("h3",{id:"_4-2-算法实现",tabindex:"-1"},[m("a",{class:"header-anchor",href:"#_4-2-算法实现"},[m("span",null,"4.2 算法实现")])],-1),m("p",null,[s("对于方程"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"a"),m("mi",null,"x"),m("mo",null,"+"),m("mi",null,"b"),m("mi",null,"y"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,","),m("mi",null,"b"),m("mo",{stretchy:"false"},")")])])]),s(", 假设方程"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mi",null,"b"),m("mi",null,"x"),m("mo",null,"+"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mi",null,"y"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,","),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"b"),m("mo",{stretchy:"false"},")")])])]),s("有整数解"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-msup",{space:"2"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.363em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"x"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,","),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",{stretchy:"false"},")")])])]),s("，由于"),m("mjx-container",{class:"MathJax",jax:"CHTML",style:{position:"relative"}},[m("mjx-math",{class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c67"}),m("mjx-c",{class:"mjx-c63"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"inline"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",null,","),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("mo",null,"="),m("mo",{"data-mjx-texclass":"OP",movablelimits:"true"},"gcd"),m("mo",{stretchy:"false"},"("),m("mi",null,"b"),m("mo",null,","),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"b"),m("mo",{stretchy:"false"},")")])])]),s(",可以得到")],-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4",rspace:"4"},[m("mjx-c",{class:"mjx-c6D"}),m("mjx-c",{class:"mjx-c6F"}),m("mjx-c",{class:"mjx-c64"})]),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-mi",{class:"mjx-i",space:"4"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2B"})]),m("mjx-mi",{class:"mjx-i",space:"3"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c28"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c230A"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c230B"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c29"})])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"a"),m("mi",null,"x"),m("mo",null,"+"),m("mi",null,"b"),m("mi",null,"y"),m("mo",null,"="),m("mi",null,"b"),m("msup",null,[m("mi",null,"x"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,"+"),m("mo",{stretchy:"false"},"("),m("mi",null,"a"),m("mo",{lspace:"thickmathspace",rspace:"thickmathspace"},"mod"),m("mi",null,"b"),m("mo",{stretchy:"false"},")"),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,"="),m("mi",null,"a"),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,"+"),m("mi",null,"b"),m("mo",{stretchy:"false"},"("),m("msup",null,[m("mi",null,"x"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,"−"),m("mo",{fence:"false",stretchy:"false"},"⌊"),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"b")]),m("mo",{fence:"false",stretchy:"false"},"⌋"),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",{stretchy:"false"},")")])])],-1),m("p",null,"所以",-1),m("mjx-container",{class:"MathJax",jax:"CHTML",display:"true",style:{position:"relative"}},[m("mjx-math",{display:"true",style:{"margin-left":"0","margin-right":"0"},class:"MJX-TEX","aria-hidden":"true"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c2C"})]),m("mjx-mi",{class:"mjx-i",space:"2"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-mo",{class:"mjx-n",space:"4"},[m("mjx-c",{class:"mjx-c3D"})]),m("mjx-msup",{space:"4"},[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D465 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c2212"})]),m("mjx-mo",{class:"mjx-n",space:"3"},[m("mjx-c",{class:"mjx-c230A"})]),m("mjx-mfrac",null,[m("mjx-frac",{type:"d"},[m("mjx-num",null,[m("mjx-nstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44E TEX-I"})])]),m("mjx-dbox",null,[m("mjx-dtable",null,[m("mjx-line",{type:"d"}),m("mjx-row",null,[m("mjx-den",null,[m("mjx-dstrut",{type:"d"}),m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D44F TEX-I"})])])])])])])]),m("mjx-mo",{class:"mjx-n"},[m("mjx-c",{class:"mjx-c230B"})]),m("mjx-msup",null,[m("mjx-mi",{class:"mjx-i"},[m("mjx-c",{class:"mjx-c1D466 TEX-I"})]),m("mjx-script",{style:{"vertical-align":"0.413em"}},[m("mjx-mo",{class:"mjx-var",size:"s"},[m("mjx-c",{class:"mjx-c2032"})])])])]),m("mjx-assistive-mml",{unselectable:"on",display:"block"},[m("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[m("mi",null,"x"),m("mo",null,"="),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,","),m("mi",null,"y"),m("mo",null,"="),m("msup",null,[m("mi",null,"x"),m("mo",{"data-mjx-alternate":"1"},"′")]),m("mo",null,"−"),m("mo",{fence:"false",stretchy:"false"},"⌊"),m("mfrac",null,[m("mi",null,"a"),m("mi",null,"b")]),m("mo",{fence:"false",stretchy:"false"},"⌋"),m("msup",null,[m("mi",null,"y"),m("mo",{"data-mjx-alternate":"1"},"′")])])])],-1),l(`<p>用递归算法实现</p><div class="language-cpp" data-highlighter="prismjs" data-ext="cpp" data-title="cpp"><pre><code><span class="line">pair<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token punctuation">,</span><span class="token keyword">int</span><span class="token operator">&gt;</span> <span class="token function">exgcd</span><span class="token punctuation">(</span><span class="token keyword">int</span> a<span class="token punctuation">,</span><span class="token keyword">int</span> b<span class="token punctuation">)</span><span class="token punctuation">{</span></span>
<span class="line">    <span class="token keyword">if</span><span class="token punctuation">(</span>b <span class="token operator">==</span> <span class="token number">0</span><span class="token punctuation">)</span><span class="token keyword">return</span> <span class="token function">make_pair</span><span class="token punctuation">(</span><span class="token number">1</span><span class="token punctuation">,</span><span class="token number">0</span><span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line">    pair<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token punctuation">,</span><span class="token keyword">int</span><span class="token operator">&gt;</span>temp <span class="token operator">=</span> <span class="token function">exgcd</span><span class="token punctuation">(</span>b<span class="token punctuation">,</span>a<span class="token operator">%</span>b<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line">    pair<span class="token operator">&lt;</span><span class="token keyword">int</span><span class="token punctuation">,</span><span class="token keyword">int</span><span class="token operator">&gt;</span>ans <span class="token operator">=</span> <span class="token function">make_pair</span><span class="token punctuation">(</span>temp<span class="token punctuation">.</span>second<span class="token punctuation">,</span>temp<span class="token punctuation">.</span>first<span class="token operator">-</span><span class="token punctuation">(</span><span class="token keyword">int</span><span class="token punctuation">)</span><span class="token punctuation">(</span>a<span class="token operator">/</span>b<span class="token punctuation">)</span><span class="token operator">*</span>temp<span class="token punctuation">.</span>second<span class="token punctuation">)</span><span class="token punctuation">;</span></span>
<span class="line">    <span class="token keyword">return</span> ans<span class="token punctuation">;</span></span>
<span class="line"><span class="token punctuation">}</span></span>
<span class="line"></span></code></pre></div>`,2)]))}const r=a(t,[["render",i],["__file","number_theory_1.html.vue"]]),p=JSON.parse('{"path":"/note/math/number_theory_1.html","title":"数论（一）","lang":"zh-CN","frontmatter":{},"headers":[{"level":2,"title":"1.同余系统","slug":"_1-同余系统","link":"#_1-同余系统","children":[{"level":3,"title":"1.1 最大公约数(Greatest common divisor)","slug":"_1-1-最大公约数-greatest-common-divisor","link":"#_1-1-最大公约数-greatest-common-divisor","children":[]},{"level":3,"title":"1.2 最小公倍数(Least Common Multiple)","slug":"_1-2-最小公倍数-least-common-multiple","link":"#_1-2-最小公倍数-least-common-multiple","children":[]},{"level":3,"title":"1.3 模运算","slug":"_1-3-模运算","link":"#_1-3-模运算","children":[]},{"level":3,"title":"1.4 同余等式","slug":"_1-4-同余等式","link":"#_1-4-同余等式","children":[]}]},{"level":2,"title":"2.中国剩余定理","slug":"_2-中国剩余定理","link":"#_2-中国剩余定理","children":[{"level":3,"title":"2.1 孙子算经中的问题","slug":"_2-1-孙子算经中的问题","link":"#_2-1-孙子算经中的问题","children":[]},{"level":3,"title":"2.2 数学解释","slug":"_2-2-数学解释","link":"#_2-2-数学解释","children":[]},{"level":3,"title":"2.3 直觉","slug":"_2-3-直觉","link":"#_2-3-直觉","children":[]},{"level":3,"title":"2.4 数学表达","slug":"_2-4-数学表达","link":"#_2-4-数学表达","children":[]},{"level":3,"title":"2.5 举例，孙子算经中的问题","slug":"_2-5-举例-孙子算经中的问题","link":"#_2-5-举例-孙子算经中的问题","children":[]}]},{"level":2,"title":"3.欧拉定理","slug":"_3-欧拉定理","link":"#_3-欧拉定理","children":[{"level":3,"title":"3.1 欧拉函数","slug":"_3-1-欧拉函数","link":"#_3-1-欧拉函数","children":[]},{"level":3,"title":"3.2 欧拉定理","slug":"_3-2-欧拉定理","link":"#_3-2-欧拉定理","children":[]},{"level":3,"title":"3.3 模反元素（逆元）","slug":"_3-3-模反元素-逆元","link":"#_3-3-模反元素-逆元","children":[]},{"level":3,"title":"3.4 费马小定理","slug":"_3-4-费马小定理","link":"#_3-4-费马小定理","children":[]}]},{"level":2,"title":"4.扩展碾压相除法","slug":"_4-扩展碾压相除法","link":"#_4-扩展碾压相除法","children":[{"level":3,"title":"4.1 裴蜀定理","slug":"_4-1-裴蜀定理","link":"#_4-1-裴蜀定理","children":[]},{"level":3,"title":"4.2 算法实现","slug":"_4-2-算法实现","link":"#_4-2-算法实现","children":[]}]}],"git":{"updatedTime":1740235322000,"contributors":[{"name":"thejinchao","username":"thejinchao","email":"thejinchao@gmail.com","commits":1,"url":"https://github.com/thejinchao"}]},"filePathRelative":"note/math/number_theory_1.md"}');export{r as comp,p as data};