题目
总算遇到一题不毒瘤的山东省选题了
做法
(prodlimits_{i=1}^nprodlimits_{j=1}^mf[gcd(i,j)])
根据套路枚举(gcd)
(prodlimits_{g=1}^{min(n,m)}prodlimits_{i=1}^nprodlimits_{j=1}[gcd(i,j)=d]f[d])
(prodlimits_{g=1}^{min(n,m)}f[g]^{ sumlimits_{i=1}^{frac{n}{d}} sumlimits_{j=1}^{frac{m}{d}} [gcd(i,j)=1] })
反演(sumlimits_{i=1}^{frac{n}{d}} sumlimits_{j=1}^{frac{m}{d}} [gcd(i,j)=1]):(sumlimits_{i=1}^{frac{n}{d}}mu(i)lfloor frac{n}{id} floor lfloor frac{m}{id} floor)
(prodlimits_{d=1}^{min(n,m)}f[d]^{ sumlimits_{i=1}^{frac{n}{d}}mu(i)lfloor frac{n}{id} floor lfloor frac{m}{id} floor })
枚举(T=id):(prod_{T=1}^{min(n,m)}(prod_{d|T}f[d]^{mu(T/d)})^{[n/T][m/T]})
My complete code
#include<cstdio>
#include<cstring>
#include<iostream>
#include<algorithm>
using namespace std;
typedef long long LL;
const LL p=1e9+7;
const int maxn=10000000;
inline int Read(){
int x(0),f(1); char c=getchar();
while(c<'0'||c>'9'){
if(c=='-')f=-1; c=getchar();
}
while(c>='0'&&c<='9')
x=(x<<3)+(x<<1)+c-'0',c=getchar();
return x*f;
}
inline LL Pow(LL base,LL b){
LL ret(1);
while(b){
if(b&1)
ret=ret*base%p;
base=base*base%p;
b>>=1;
}
return ret;
}
int mu[maxn],prime[maxn];
LL f[maxn],g[maxn],F[maxn],sum[maxn];
bool visit[maxn];
inline void F_phi(int N){
mu[1]=1;
int tot(0);
f[1]=1;
for(int i=2;i<=N;++i){
f[i]=(f[i-1]+f[i-2])%p,
g[i]=Pow(f[i],p-2),
F[i]=1;
if(!visit[i]){
prime[++tot]=i,
mu[i]=-1;
}
for(int j=1;j<=tot&&i*prime[j]<=N;++j){
visit[i*prime[j]]=true;
if(i%prime[j]==0)
break;
else
mu[i*prime[j]]=-mu[i];
}
}
g[1]=F[1]=F[0]=1;
for(int i=1;i<=N;++i){
if(!mu[i])
continue;
for(int d=1;d*i<=N;++d)
F[i*d]=1ll*F[i*d]*(mu[i]==1?f[d]:g[d])%p;
}
sum[0]=1ll;
for(int i=1;i<=N;++i)
sum[i]=sum[i-1]*F[i]%p;
}
int T;
int main(){
F_phi(1000000);
T=Read();
while(T--){
int n(Read()),m(Read());
if(n>m)
swap(n,m);
LL ans(1);
for(int l=1,r;l<=n;l=r+1){
r=min(n/(n/l),m/(m/l));
LL ret=sum[r]*Pow(sum[l-1],p-2)%p;
ans=(ans*Pow(ret,1ll*(n/l)*(m/l)%(p-1)))%p;
}
printf("%lld
",ans);
}
return 0;
}/*
3
10000 100000
999 555
32465 485645
*/