洛谷_1445 樱花(数论)

樱花

题目链接：https://www.luogu.com.cn/problem/P1445

题解：

$\frac{1}{x}+\frac{1}{y}=\frac{1}{n!}$
$=>\frac{x+y}{xy}=\frac{1}{n!}$
$=>xy=x\ast n!+y\ast n!$
$=>y(x-n!)=x\ast n!$
$=>y=\frac{x\ast n!}{x-n!}$
令 $z=x-n!$,则$x=z+n!$
$=>y=\frac{n!\ast (z+n!)}{z}$
$=>y=n!+\frac{n!\ast n!}{z}$
$x,y$都是正整数，则$n!\ast n!$需能整除z，求$n!\ast n!$的因数数量即可。

#include
#include
#include
#include
#include
#include
#include
#include
#include
#include
#define dbg(x) cout<<#x<<" = "<
#define INF 0x3f3f3f3f
#define eps 1e-6
   
using namespace std;
typedef long long LL;   
typedef pair<int, int> P;
const int maxn = 1000100;
const int mod = 1000000007;
int a[maxn];
LL getnum(int n, int x);

int main()
{
    int n, i, j, k;
    LL ans = 1;
    scanf("%d", &n);
    for(i=2;i<=n;i++)
        if(!a[i]){
            for(j=i+i;j<=n;j+=i)
                a[j] = 1;
            LL num = 2*getnum(n, i)+1;
            ans = ans*num%mod;
        }
    printf("%lld\n", ans);
    return 0;
}

LL getnum(int n, int x)
{
    LL res = 0;
    while(n){
        res += n/x;
        n /= x;
    }
    return res;
}