Calculates the number of 0 at the end of the factorial n!

Problem description
Given the parameter n (n is a positive integer), calculate the factorial of n n! The number of "0" is included at the end.
For example, 5! =120, which contains the number of "0" at the end of the 1;10! = 3628800, with the number of "0" at the end of the 2;20! = 2432902008176640000, with the number of "0" at the end of which is 4.

Problem Analysis:

Obviously, for the factorial growth rate of very fast, it is easy to overflow. Of course, it is impossible to calculate the factorial, but to find a regular solution. The following with factoring thinking to consider: The end of the 0 can be decomposed into 2*5, a 5, a 2 corresponds to a 0;

The following is a recursive process:

(1) When n<5, f (n) = 0; Conclusion established

(2) When n>=5, can make n! =[5k*5 (k-1) ... 10*5]*a, where N=5k+r, R (0<=r<5), A is an integer without a factor, can be seen as an integer satisfying the condition;

For sequence 5k,5 (k-1) ... 10*5 in each 5i, there is an even number within the interval (5 (i-1), 5i), i.e. there is a 2 corresponding to it. So here's K ' 5 ' factor with n! The end of the 0-digit number is one by one corresponding.

So the recursive formula translates to:

F (n!) =g (5^k * k! *a) = k + g (k!) = K+f (k!); K! is the result of multiplying the coefficients.

F (n!) =k+f (k!);

Example: F (5!) = 1+f (1!) = 1;

F (10!) =2+f (2!) = 2;

Here are two implementations of C + + code: recursive; non-recursive:

recursive int n_jie_tail_zeors_recurise (int n) {    int zeros_num =0;    int k;    if (n < 5)        return 0;    for (k=5;k<=n;k+=5)    {        zeros_num++;    }    return zeros_num + n_jie_tail_zeors_recurise (zeros_num);}

Non-recursive int n_jie_tail_zeros_non_recurse (int n) {    int zeros_num =0;    int temp = n;    int k=0;    int index=0;    while (temp >5)    {        index =0;        for (k =5;k<=temp;k+=5)        {            zeros_num++;            index++;        }        temp = index;    }    return zeros_num;}

Calculates the number of 0 at the end of the factorial n!