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Sol

题目要求\(\sum_{i=1}^{n!}[gcd(i, m!)==1]\)

设\(N=n!，M=m!\)，莫比乌斯反演一波

就变成了\(\sum_{d|M}\mu(d)\frac{N}{d}\)

因为\(M|N\)所以\(d|N\)

而有个定理\(\sum_{d|M}\frac{\mu(d)}{d}=\frac{\varphi(M)}{M}\)

那么就是求\(\frac{\varphi(M)}{M}*N\)

就是\(\varphi(m!)*\frac{n!}{m!}\)

而\(\varphi(m!)=\varphi(m)*(m-1)!\)

化简

\[ans=n!*\Pi_{P|m}(1-\frac{1}{P}) \ \ \ \ (P为质数) \\
=n!*\Pi_{P|m}\frac{P-1}{P}
\]

那就变成SB题了

预处理就好了

# include <bits/stdc++.h>
# define IL inline
# define RG register
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long ll;
const int _(1e7 + 1);IL ll Read(){
RG char c = getchar(); RG ll x = 0, z = 1;
for(; c < '0' || c > '9'; c = getchar()) z = c == '-' ? -1 : 1;
for(; c >= '0' && c <= '9'; c = getchar()) x = (x << 1) + (x << 3) + (c ^ 48);
return x * z;
}int n, m, Zsy, prime[_], num, fac[_], inv[_], id[_];
bool isprime[_];IL int Pow(RG ll x, RG ll y){
RG ll ret = 1;
for(; y; y >>= 1, x = x * x % Zsy) if(y & 1) ret = ret * x % Zsy;
return ret;
}IL void Sieve(){
isprime[1] = 1; fac[1] = 1;
for(RG int i = 2; i < _; ++i){
if(!isprime[i]) prime[++num] = i , inv[num] = Pow(i, Zsy - 2);
for(RG int j = 1; j <= num && i * prime[j] < _; ++j){
isprime[i * prime[j]] = 1;
if(!(i % prime[j])) break;
}
fac[i] = 1LL * fac[i - 1] * i % Zsy;
}
for(RG int i = 1; i < num; ++i)
for(RG int j = prime[i]; j < prime[i + 1]; ++j) id[j] = i;
inv[0] = prime[0] = 1;
for(RG int i = 1; i <= num; ++i){
prime[i] = 1LL * (prime[i] - 1) * prime[i - 1] % Zsy;
inv[i] = 1LL * inv[i] * inv[i - 1] % Zsy;
}
}IL int Calc(){  return 1LL * fac[n] * prime[id[m]] % Zsy * inv[id[m]] % Zsy; }int main(RG int argc, RG char* argv[]){
RG int T = Read(); Zsy = Read();
Sieve();
while(T--){
n = Read(); m = Read();
printf("%d\n", Calc());
}
return 0;
}