10202. 「一本通 6.2 练习 5」樱花

题意

求不定方程:
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{n!}
$$
的正整数解 $(x,y)$ 的数目。

思路

$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{n!}
$$

$$
\frac{y}{xy}+\frac{x}{xy}=\frac{1}{n!}
$$

$$
\frac{x+y}{xy}=\frac{1}{n!}
$$

$$
n!\times(x+y)=xy
$$

$$
x+y=\frac{xy}{n!}
$$

$$
(x-n!)*(y-n!)=(n!)^2
$$

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#include<bits/stdc++.h>
using namespace std;
long long f[1000010],v[1000010],tot,ans[1000010],Ans=0;
long long n;
void prime(){
for(long long i=2;i<=1000000;i++){
if(!v[i]) v[i]=i,f[++tot]=i;
for(long long j=1;j<=tot;j++){
if(f[j]>v[i]||f[j]>1000000/i) break;
v[i*f[j]]=f[j];
}
}
}
int main(){
scanf("%lld",&n);
prime();long long tmp=n;
memset(ans,0,sizeof(ans));Ans=1;
for(int i=1;f[i]<=n&&i<=tot;i++){
long long tmp=0;
for(long long j=f[i];j<=n;j*=f[i]){
tmp+=n/j;
tmp%=1000000007;
}
Ans*=2*tmp+1;
Ans%=1000000007;
}
printf("%lld\n",Ans);
}
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