2019 Multi-University Training Contest 3 T6 - Fansblog
Fansblog
Time Limit: 2000/2000 MS (Java/Others) Memory Limit: 65536/65536 K (Java/Others)Total Submission(s): 374 Accepted Submission(s): 107
Problem Description
Farmer John keeps a website called ‘FansBlog’ .Everyday , there are many people visited this blog.One day, he find the visits has reached P , which is a prime number.He thinks it is a interesting fact.And he remembers that the visits had reached another prime number.He try to find out the largest prime number Q ( Q < P ) ,and get the answer of Q! Module P.But he is too busy to find out the answer. So he ask you for help. ( Q! is the product of all positive integers less than or equal to n: n! = n * (n-1) * (n-2) * (n-3) *… * 3 * 2 * 1 . For example, 4! = 4 * 3 * 2 * 1 = 24 )
Input
First line contains an number T(1<=T<=10) indicating the number of testcases.
Then T line follows, each contains a positive prime number P (1e9≤p≤1e14)
Output
For each testcase, output an integer representing the factorial of Q modulo P.
Sample Input
1
1000000007
Sample Output
328400734
题意
给出一个质数p,每一次询问\(s!\%p,(s\text{为小于p的最大质数})\)。
题解
定理:\((p-1)!\equiv p-1 \space(\mod p)\),p 为质数。
并且,质数以ln分配。
所以,$ans \sum_{i=s+1}^{p-1}i\equiv p-1(\mod p) $
所以,$ ans\equiv p-1\sum_{i=s+1}{p-1}i{-1}(\mod p) $
代码
#include<bits/stdc++.h>
#define int long long
using namespace std;
int prime[10]={2,3,5,7,11,13,19,61,2333,24251};
long long M;
int Quick_Multiply(int a,int b,int c)
{
long long ans=0,res=a;
while(b)
{
if(b&1)
ans=(ans+res)%c;
res=(res+res)%c;
b>>=1;
}
return (int)ans;
}
int Quick_Power(int a,int b,int c)
{
int ans=1,res=a;
while(b)
{
if(b&1)
ans=Quick_Multiply(ans,res,c);
res=Quick_Multiply(res,res,c);
b>>=1;
}
return ans;
}
bool Miller_Rabin(int x)
{
int i,j,k;
int s=0,t=x-1;
if(x==2) return true;
if(x<2||!(x&1)) return false;
while(!(t&1))
{
s++;
t>>=1;
}
for(i=0;i<10&&prime[i]<x;++i)
{
int a=prime[i];
int b=Quick_Power(a,t,x);
for(j=1;j<=s;++j)
{
k=Quick_Multiply(b,b,x);
if(k==1&&b!=1&&b!=x-1)
return false;
b=k;
}
if(b!=1) return false;
}
return true;
}
signed main()
{
int T;
cin >> T;
while (T--){
int x;
int ans;
scanf("%lld",&x);
ans = x - 1;
int M = x;
while (Miller_Rabin(x-1) == 0) x--, ans = Quick_Multiply(ans, Quick_Power(x,M-2,M),M);
cout << ans << endl;
}
return 0;
}
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