【LG1445】樱花

【LG1445】樱花

题面

洛谷

题解

\[\frac 1x+\frac 1y=\frac 1{n!}\\ \frac{x+y}{xy}=\frac 1{n!}\\ n!(x+y)=xy\\ xy-n!(x+y)=0\\ (n!)^2+xy-n!(x+y)=(n!)^2\\ (x-n!)(y-n!)=(n!)^2 \]

设\(a=x-n!,b=y-n!\)

\[ab=(n!)^2 \]

枚举约数就好啦

代码

#include <iostream>
#include <cstdio>
#include <cstdlib>
#include <cstring> 
#include <cmath> 
#include <algorithm>
using namespace std;  
const int Mod = 1e9 + 7;
const int MAX_N = 1e6 + 5; 
void pls(int &x, int y) { x += y; if (x >= Mod) x -= Mod; } 
bool is_prime[MAX_N]; 
int N = 1e6, prime[MAX_N], num, ans = 1; 
void sieve() { 
	for (int i = 1; i <= N; i++) is_prime[i] = 1; 
	is_prime[1] = 0; 
	for (int i = 2; i <= N; i++) { 
		if (is_prime[i]) prime[++num] = i; 
		for (int j = 1; prime[j] * i <= N && j <= num; j++) { 
			is_prime[i * prime[j]] = 0;
			if (!(i % prime[j])) break; 
		} 
	} 
} 
int main () {
	sieve(); 
	cin >> N; 
	for (int i = 1; i <= num; i++) {
		if (prime[i] > N) break;
		int cnt = 0; 
		for (long long j = prime[i]; j <= N; j = j * prime[i]) pls(cnt, 1ll * N / j);
		pls(cnt, cnt), pls(cnt, 1); 
		ans = 1ll * ans * cnt % Mod; 
	} 
	printf("%d\n", ans); 
	return 0; 
}