grandson定理

用处：求解同余线性方程组

inv：逆元

一堆物品

3个3个分剩2个

5个5个分剩3个

7个7个分剩2个

问这个物品有多少个

5*7*inv(5*7,  3) % 3  =  1

3*7*inv(3*7,  5) % 5  =  1

3*5*inv(3*5,  7) % 7  =  1

然后两边同乘你需要的数

2 * 5*7*inv(5*7,  3) % 3  =  2

3 * 3*7*inv(3*7,  5) % 5  =  3

2 * 3*5*inv(3*5,  7) % 7  =  2

令

a = 2 * 5*7*inv(5*7,  3)     b = 3 * 3*7*inv(3*7,  5)     c = 2 * 3*5*inv(3*5,  7)

那么

a % 3 = 2    b % 5 = 3    c % 7 = 2  答案就是a+b+c

因为

a%5 = a%7 = 0 因为a是5的倍数，也是7的倍数    b%3 = b%7 = 0 因为b是3的倍数，也是7的倍数    c%3 = c%5 = 0 因为c是3的倍数，也是5的倍数

所以

(a + b + c) % 3 = (a % 3) + (b % 3) + (c % 3) = 2 + 0 + 0 = 2

(a + b + c) % 5 = (a % 5) + (b % 5) + (c % 5) = 0 + 3 + 0 = 3

(a + b + c) % 7 = (a % 7) + (b % 7) + (c % 7) = 0 + 0 + 2 = 2

每105个就是一个答案（105 = 3 * 5 * 7）

根据计算，最小的答案等于233,233%105 = 23

//n个方程：x=a[i](mod m[i]) (0<=i<n)
LL china(int n, LL *a, LL *m){
    LL M = , ret = ;
    for(int i = ; i < n; i ++) M *= m[i];
    for(int i = ; i < n; i ++){
        LL w = M / m[i];
        ret = (ret + w * inv(w, m[i]) * a[i]) % M;
    }
    return (ret + M) % M;
}

两两互质代码

例题：poj 1006

人自出生起就有体力，情感和智力三个生理周期，分别为23，28和33天。一个周期内有一天为峰值，在这一天，人在对应的方面（体力，情感或智力）表现最好。通常这三个周期的峰值不会是同一天。现在给出三个日期，分别对应于体力，情感，智力出现峰值的日期。然后再给出一个起始日期，要求从这一天开始，算出最少再过多少天后三个峰值同时出现。

#include<cstdio>
typedef long long LL;
const int N =  + ;
void ex_gcd(LL a, LL b, LL &x, LL &y, LL &d){
    if (!b) {d = a, x = , y = ;}
    else{
        ex_gcd(b, a % b, y, x, d);
        y -= x * (a / b);
    }
}
LL inv(LL t, LL p){//如果不存在，返回-1
    LL d, x, y;
    ex_gcd(t, p, x, y, d);
    return d ==  ? (x % p + p) % p : -;
}
LL china(int n, LL *a, LL *m){//中国剩余定理
    LL M = , ret = ;
    for(int i = ; i < n; i ++) M *= m[i];
    for(int i = ; i < n; i ++){
        LL w = M / m[i];
        ret = (ret + w * inv(w, m[i]) * a[i]) % M;
    }
    return (ret + M) % M;
}
int main(){
    LL p[], r[], d, ans, MOD = ;
    int cas = ;
    p[] = ; p[] = ; p[] = ;
    while(~scanf("%I64d%I64d%I64d%I64d", &r[], &r[], &r[], &d) && (~r[] || ~r[] || ~r[] || ~d)){
        ans = ((china(, r, p) - d) % MOD + MOD) % MOD;
        printf("Case %d: the next triple peak occurs in %I64d days.\n", ++cas, ans ? ans : );
    }

}

solve one

如果两两不保证互质的话

#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long LL;
typedef pair<LL, LL> PLL;
PLL linear(LL A[], LL B[], LL M[], int n) {//求解A[i]x = B[i] (mod M[i]),总共n个线性方程组
    LL x = , m = ;
    for(int i = ; i < n; i ++) {
        LL a = A[i] * m, b = B[i] - A[i]*x, d = gcd(M[i], a);
        if(b % d != )  return PLL(, -);//答案不存在，返回-1
        LL t = b/d * inv(a/d, M[i]/d)%(M[i]/d);
        x = x + m*t;
        m *= M[i]/d;
    }
    x = (x % m + m ) % m;
    return PLL(x, m);//返回的x就是答案，m是最后的lcm值
}

两两不互质

例题：poj 2891

给出k个模方程组：x mod ai = ri。求x的最小正值。如果不存在这样的x，那么输出-1.

#include<cstdio>
#include<algorithm>
using namespace std;
typedef long long LL;
typedef pair<LL, LL> PLL;
LL a[], b[], m[];
LL gcd(LL a, LL b){
    return b ? gcd(b, a%b) : a;
}
void ex_gcd(LL a, LL b, LL &x, LL &y, LL &d){
    if (!b) {d = a, x = , y = ;}
    else{
        ex_gcd(b, a % b, y, x, d);
        y -= x * (a / b);
    }
}
LL inv(LL t, LL p){//如果不存在，返回-1
    LL d, x, y;
    ex_gcd(t, p, x, y, d);
    return d ==  ? (x % p + p) % p : -;
}
PLL linear(LL A[], LL B[], LL M[], int n) {//求解A[i]x = B[i] (mod M[i]),总共n个线性方程组
    LL x = , m = ;
    for(int i = ; i < n; i ++) {
        LL a = A[i] * m, b = B[i] - A[i]*x, d = gcd(M[i], a);
        if(b % d != )  return PLL(, -);//答案，不存在，返回-1
        LL t = b/d * inv(a/d, M[i]/d)%(M[i]/d);
        x = x + m*t;
        m *= M[i]/d;
    }
    x = (x % m + m ) % m;
    return PLL(x, m);//返回的x就是答案，m是最后的lcm值
}
int main(){
    int n;
    while(scanf("%d", &n) != EOF){
        for(int i = ; i < n; i ++){
            a[i] = ;
            scanf("%d%d", &m[i], &b[i]);
        }
        PLL ans = linear(a, b, m, n);
        if(ans.second == -) printf("-1\n");
        else printf("%I64d\n", ans.first);
    }
}

solve two