求C(n, m) % (p1*p2*p3*...*pk)的值
范围:1≤m≤n≤1e18,1≤k≤10,pi≤1e5,保证p1*p2*p3*...*pk≤1e18
#include<cstdio>
typedef long long LL;
const int N = 100000 + 5;
LL mul(LL a, LL b, LL p){//快速乘,计算a*b%p
LL ret = 0;
while(b){
if(b & 1) ret = (ret + a) % p;
a = (a + a) % p;
b >>= 1;
}
return ret;
}
LL fact(int n, LL p){//n的阶乘求余p
LL ret = 1;
for (int i = 1; i <= n ; i ++) ret = ret * i % p ;
return ret ;
}
void ex_gcd(LL a, LL b, LL &x, LL &y, LL &d){
if (!b) {d = a, x = 1, y = 0;}
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 == 1 ? (x % p + p) % p : -1;
}
LL comb(int n, int m, LL p){//C(n, m) % p
if (m < 0 || m > n) return 0;
return fact(n, p) * inv(fact(m, p), p) % p * inv(fact(n-m, p), p) % p;
}
LL Lucas(LL n, LL m, int p){
return m ? Lucas(n/p, m/p, p) * comb(n%p, m%p, p) % p : 1;
}
LL china(int n, LL *a, LL *m){//中国剩余定理
LL M = 1, ret = 0;
for(int i = 0; i < n; i ++) M *= m[i];
for(int i = 0; i < n; i ++){
LL w = M / m[i];
//ret = (ret + w * inv(w, m[i]) * a[i]) % M;//这句写了会WA,用下面那句
ret = (ret + mul(w * inv(w, m[i]), a[i], M)) % M;
//因为这里直接乘会爆long long ,所以我用快速乘(unsigned long long也是爆掉,除非用高精度)
}
return (ret + M) % M;
}
int main(){
int T, k;
LL n, m, p[15], r[15];
scanf("%d", &T);
while(T--){
scanf("%I64d%I64d%d", &n, &m, &k);
for(int i = 0; i < k; i ++){
scanf("%I64d", &p[i]);
r[i] = Lucas(n, m, p[i]);
}
printf("%I64d\n", china(k, r, p));
}
}