uva 11440 - Help Tomisu(欧拉函数)

题目链接：uva 11440 - Help Tomisu

题目大意：给定n和m，求从2~n！中的数x，要求x的质因子均大于m，问说x有多少个，答案模上1e9+7。

解题思路：

(1)n!=k∗m!(n≥m)

(2) 如果有gcd(x,T)=1，那么gcd(x+T,T)=gcd(x,T)=1

题目要求说x的质因子必须要大于m，也就是说x不能包含2~m的因子，那么gcd(x,m!)=1,于是我们求出ϕ(m!),小于m！并且满足gcd(x,m!)=1的个数。

那么根据(2)可得从[m!+1, 2*m!]中的x个数也是ϕ(m!)个；因为如果存在gcd(x,T)=a,那么gcd(x+T,T)=gcd(x,T)=a.

又因为(1),所以最后n!以内的x个数为：n!∗ϕ(m!)m!

#include <cstdio>#include <cstring>#include <cmath>#include <algorithm>using namespace std;typedef long long ll;const int maxn = 1e7;const ll MOD = 100000007;int np, pri[maxn+5], vis[maxn+5];ll fact[maxn+5], phi[maxn+5];void prime_table (ll n) {    np = 0;    for (ll i = 2; i <= n; i++) {        if (vis[i])            continue;        pri[np++] = i;        for (ll j = i * i; j <= n; j += i)            vis[j] = 1;    }}void gcd (ll a, ll b, ll& d, ll& x, ll& y) {    if (b == 0) {        d = a;        x = 1;        y = 0;    } else {        gcd(b, a%b, d, y, x);        y -= (a/b) * x;    }}inline ll inv_number (ll a, ll n) {    ll d, x, y;    gcd(a, n, d, x, y);    return (x + n) % n;}void init (ll n) {    fact[1] = phi[1] = 1;    for (ll i = 2; i <= n; i++) {        fact[i] = (fact[i-1] * i) % MOD;        phi[i] = phi[i-1];        if (vis[i] == 0) {            phi[i] *= ((i-1) * inv_number(i, MOD)) % MOD;            phi[i] %= MOD;        }    }}ll solve (int n, int m) {    ll ans = fact[n] * phi[m] % MOD;    return (ans - 1 + MOD) % MOD;}int main () {    prime_table(maxn);    init (maxn);    int n, m;    while (scanf("%d%d", &n, &m) == 2 && n + m) {        printf("%lld\n", solve(n, m));    }    return 0;}