HDU 1018 Big Number 数学题解

Problem Description

In many applications very large integers numbers are required. Some of these applications are using keys for secure transmission of data, encryption, etc. In this problem you are given a number, you have to determine the number of digits in the factorial of the number.

 

Input

Input consists of several lines of integer numbers. The first line contains an integer n, which is the number of cases to be tested, followed by n lines, one integer 1 ≤ n ≤ 10⁷ on each line.

 

Output

The output contains the number of digits in the factorial of the integers appearing in the input.

 

Sample Input

2
10
20

 

Sample Output

7
19

本题就考查斯特林公式。由于斯特林公式是求解n！

的近似公式。而本题仅仅须要求解有多少位。

底层数学原理就是求一个数n的数位能够使用 digits = log10(n)

然后利用斯特林公式求出n！的近似值就能够利用log10来求得数位了。

斯特林公式百度百科有，这里不反复了。

float不能AC的时候，就使用double吧。

#include <stdio.h>
#include <math.h>
const float PI = 3.14159265358979323846f;

inline int getDigits(int n)
{
	float num = float(n);
	int ans = (int)(0.5*log10(2.0*PI*num) + num*(log(num)-1)/log(10.0)) + 1;
	return ans;
}

int main()
{
	int T, n;
	scanf("%d", &T);
	while (T--)
	{
		scanf("%d", &n);
		printf("%d
", getDigits(n));
	}
	return 0;
}