同余定理在阶乘和取余上的运用

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本文探讨了GCC编译器系统中缺失的数学运算符“！”及其在数学中的意义，即阶乘操作。通过分析给出的代码示例，详细解释了如何在计算机程序中实现对一系列阶乘进行求和并取模的算法。该算法特别关注于当n远大于m时，如何简化计算过程，避免不必要的大数运算。

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1254: GCC
Time Limit: 1 Sec Memory Limit: 128 MB
[Submit][Status][Web Board]
Description
The GNU Compiler Collection (usually shortened to GCC) is a compiler system produced by the GNU Project supporting various programming languages. But it doesn’t contains the math operator “!”.
In mathematics the symbol represents the factorial operation. The expression n! means “the product of the integers from 1 to n”. For example, 4! (read four factorial) is 4 × 3 × 2 × 1 = 24. (0! is defined as 1, which is a neutral element in multiplication, not multiplied by anything.)
We want you to help us with this formation: (0! + 1! + 2! + 3! + 4! + … + n!)%m

Input

The first line consists of an integer T, indicating the number of test cases.
Each test on a single consists of two integer n and m.

Output

Output the answer of (0! + 1! + 2! + 3! + 4! + … + n!)%m.

Constrains
0 < T <= 20
0 <= n < 10^100 (without leading zero)
0 < m < 1000000

Sample Input

1
10 861017

Sample Output

593846

HINT

Source

2009 Asia Wuhan Regional Contest Online
AC_code:
/*
只要n>m，那么m!+(m+1)!+…+n!这些项都是可以被m整除的，要对m求余，
只需要找比m小的阶乘。
so只需判断n是否大于m，若大于m，则n取m-1，若小于m则不变。
同时根据同余定理，只需算n!%m即可！
*/

#include <stdio.h>
#include <string.h>
char a[105];
int main()
{
    int T;
    scanf("%d",&T);
    while(T--)
    {
        memset(a,'\0',sizeof(a));
        int n = 0,m;
        scanf("%s %d",a,&m);
        long long sum = 1,l = strlen(a),num = 1;
        for(int i = 0; i < l; i++)
        {
            n = n*10 + (a[i]-'0');
            if(n >= m) break;
        }
        if(n >= m||l > 7) n = m-1;//当长度大于7或n>=m时只需算(m-1)!
        for(int i = 1; i <= n; i++)
        {
            num = num*i%m;
            sum = (sum+num)%m;
            if(num == 0) break;
        }
        printf("%lld\n",sum%m);//这里必须也要取余
    }
    return 0;
}