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#include <bits/stdc++.h>
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using namespace std;
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typedef long long LL;
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int a[20], m[20], n;
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//【模板】中国剩余定理
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// https://www.bilibili.com/video/av25823277
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// https://xiaoxiaoh.blog.csdn.net/article/details/103024423
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//扩展欧几里得
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LL exgcd(LL a, LL b, LL &x, LL &y) {
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if (b == 0) {
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x = 1, y = 0;
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return a;
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}
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LL d = exgcd(b, a % b, y, x);
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y -= a / b * x;
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return d;
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}
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//中国剩余定理
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LL CRT() {
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LL ans = 0;
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LL M = 1;
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LL t1, t2;
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for (int i = 1; i <= n; i++) M *= m[i]; //M:所有除数的公倍数,因为是两两互质的，所以，也是最小公倍数
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for (int i = 1; i <= n; i++) {
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LL Mi = M / m[i]; //Mi:把自己除掉后，其它几个除数的乘积
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//求逆元的过程
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//求exgcd(e, m)—>利用欧几里得算法不断递归直到x=1,y=0—>反向递归求出第一层的x和y，x即为e模m的逆元。
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exgcd(Mi, m[i], t1, t2); //t1就是逆元
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ans = (ans + a[i] * Mi % M * t1) % M; //这个就是套用推导出来的公式了
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}
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return (ans + M) % M; //保证结果为正整数
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}
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// 快乘模板
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// https://www.cnblogs.com/Fy1999/p/8908522.html
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LL qmul(LL a, LL b, LL m) {
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LL ans = 0;
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while (b) {
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if (b & 1) ans = (ans + a) % m;
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a = (a << 1) % m;
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b >>= 1;
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}
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return ans;
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}
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//中国剩余定理(快乘版本)
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LL CRT_QickMul() {
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LL ans = 0;
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LL M = 1;
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LL x, y;
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for (int i = 1; i <= n; i++) M *= m[i];
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for (int i = 1; i <= n; i++) {
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LL Mi = M / m[i];
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exgcd(Mi, m[i], x, y);
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x = (x % m[i] + m[i]) % m[i]; //变成最小正整数解,这是为了快乘的特点,转换为最小正整数
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//写一句 //d=(d%MOD+MOD)%MOD; 转化成正数就可以了,网上很多人也是这样做的,尤其在加密解密时。
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ans = (ans + qmul(a[i] * Mi % M, x , M)) % M;
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}
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return (ans + M) % M; //防止为负数
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}
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int main() {
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//输入+输出重定向
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freopen("../AcWing/N11/China.txt", "r", stdin);
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scanf("%d", &n);
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for (int i = 1; i <= n; i++) {
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scanf("%d%d", m + i, a + i); //m是除数,2是余数 ,注意这里数组的数据读入方法,还有下标的开始位置
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}
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printf("%lld\n", CRT());
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printf("%lld\n", CRT_QickMul());
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//关闭文件
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fclose(stdin);
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return 0;
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} |
