HDU5391 Zball in Tina Town 威尔逊定理

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########数论########

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威尔逊定理

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本文介绍了一个有趣的数学问题，即计算一个名为ZBall的魔法球在第(n-1)天的大小，并对其结果取模n后的值。通过分析质数与合数情况，利用威尔逊定理等数学原理给出了解决方案。

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Tina Town is a friendly place. People there care about each other.

Tina has a ball called zball. Zball is magic. It grows larger every day. On the first day, it becomes 11 time as large as its original size. On the second day,it will become 22 times as large as the size on the first day. On the n-th day,it will become nn times as large as the size on the (n-1)-th day. Tina want to know its size on the (n-1)-th day modulo n.

Input

The first line of input contains an integer TT, representing the number of cases.

The following TT lines, each line contains an integer nn, according to the description.
T≤105,2≤n≤109T≤105,2≤n≤109

Output

For each test case, output an integer representing the answer.

Sample Input

2
3
10

Sample Output

2
0

题意：

求（n-1）！%n

分析：

分合数和素数讨论，如果n是素数，根据威尔逊定理：（（p-1）！+1）%p==0，可得（p-1）！%p = p-1;

如果n是合数，当n>4时，（p-1）！一定能整除p，所以（p-1）！%p =0；特别的当 n = 1，（p-1）！%p=0；当n=4，（p-1）！%p =2；

代码：

#include <stdio.h>
using namespace std;
bool isPrime(int x){//判断x是不是质数，是返回true，不是返回false   
    if(x <= 1) return false;   
    for(int i = 2; i * i <= x; i ++){//用乘法避免根号的精度误差   
        if(x % i == 0) return false;  
    }  
    return true;  
}  
int main(){
	int t;
	scanf("%d",&t);
	while(t--){
		int n;
		scanf("%d",&n);
		if(n==4) printf("2\n");
		else if(isPrime(n)) printf("%d\n",n-1);
		else printf("0\n");
	}
}