Número final de ceros en el producto de dos factoriales

Dados dos enteros N o M, ¿encontrar el número de ceros finales en el producto de factoriales (N!*M!)? 
Ejemplos: 
 

Input : N = 4, M = 5 
Output :  1
Explanation : 4! = 24, 5! = 120
Product has only 1 trailing 0.

Input : N = 127!, M = 57!
Output : 44

Como se discutió en número de ceros en N! se puede calcular dividiendo recursivamente N por 5 y sumando los cocientes.
 

Por ejemplo, si N = 127, entonces 
¡Número de 0 en 127! = 127/5 + 127/25 + 127/125 + 127/625 
= 25 + 5 + 1 + 0 
= 31

Número de ceros en N! = 31. De manera similar, para M podemos calcular y sumar ambos. 
Por lo anterior, podemos concluir que el número de ceros en N!*M! ¡Es igual a la suma del número de ceros en N! ¡y M!.
 

f(N) = piso(N/5) + piso(N/5^2) + … piso(N/5^3) + … 
f(M) = piso(x/5) + piso(M/5^ 2) +… suelo(M/5^3) +…
Entonces la respuesta es f(N)+f(M)

// CPP program for count number of trailing zeros
// in N! * M!
#include <iostream>
using namespace std;
 
// Returns number of zeros in factorial n
int trailingZero(int x)
{
    // dividing x by powers of 5 and update count
    int i = 5, count = 0;
    while (x > i) {
        count = count + x / i;
        i = i * 5;
    }
    return count;
}
 
// Returns count of trailing zeros in M! x N!
int countProductTrailing(int M, int N)
{
   return trailingZero(N) + trailingZero(M);
}
 
// Driver program
int main()
{
    int N = 67, M = 98;
    cout << countProductTrailing(N, M);
    return 0;
}

// Java program for count number
// of trailing zeros in N! * M!
import java.io.*;
 
class GFG {
     
    // Returns number of zeros in factorial n
    static int trailingZero(int x)
    {
        // dividing x by powers
        // of 5 and update count
        int i = 5, count = 0;
         
        while (x > i) {
             
            count = count + x / i;
            i = i * 5;
        }
        return count;
    }
     
    // Returns count of trailing zeros in M! x N!
    static int countProductTrailing(int M, int N)
    {
    return trailingZero(N) + trailingZero(M);
    }
     
    // Driver program
    public static void main(String args[])
    {
        int N = 67, M = 98;
        System.out.println(countProductTrailing(N, M));
    }
}
 
 
/* This code is contributed by Nikita Tiwari.*/

# Python 3 program for count
# number of trailing zeros
# in N ! * M !
 
# Returns number of zeros in
# factorial n
def trailingZero(x) :
 
    # Dividing x by powers of 5
    # and update count
    i = 5
    count = 0
     
    while (x > i) :
        count = count + x // i
        i = i * 5
     
    return count
     
# Returns count of trailing
# zeros in M ! x N !
def countProductTrailing(M, N) :
    return trailingZero(N) + trailingZero(M)
     
# Driver program
N = 67
M = 98
print(countProductTrailing(N, M))
 
# This code is contributed by Nikita Tiwari.

// Java program for count number
// of trailing zeros in N! * M!
using System;
 
class GFG {
     
    // Returns number of zeros in factorial n
    static int trailingZero(int x)
    {
        // dividing x by powers
        // of 5 and update count
        int i = 5, count = 0;
         
        while (x > i) {
             
            count = count + x / i;
            i = i * 5;
        }
        return count;
    }
     
    // Returns count of trailing zeros in M! x N!
    static int countProductTrailing(int M, int N)
    {
    return trailingZero(N) + trailingZero(M);
    }
     
    // Driver program
    public static void Main()
    {
        int N = 67, M = 98;
        Console.WriteLine(countProductTrailing(N, M));
    }
}
 
 
/* This code is contributed by vt_m.*/

<?php
// PHP program for count number
// of trailing zeros in N! * M!
 
// Returns number of
// zeros in factorial n
function trailingZero($x)
{
     
    // dividing x by powers of
    // 5 and update count
    $i = 5; $count = 0;
    while ($x > $i)
    {
        $count = $count + (int)($x / $i);
        $i = $i * 5;
    }
    return $count;
}
 
// Returns count of trailing
// zeros in M! x N!
function countProductTrailing($M, $N)
{
    return trailingZero($N) + trailingZero($M);
}
 
// Driver Code
$N = 67; $M = 98;
echo(countProductTrailing($N, $M));
 
// This code is contributed by Ajit.
?>

<script>
// Javascript program for count number
// of trailing zeros in N! * M!
 
// Returns number of
// zeros in factorial n
function trailingZero(x)
{
     
    // dividing x by powers of
    // 5 and update count
    let i = 5;
    let count = 0;
    while (x > i)
    {
        count = count + parseInt(x / i);
        i = i * 5;
    }
    return count;
}
 
// Returns count of trailing
// zeros in M! x N!
function countProductTrailing(M, N)
{
    return trailingZero(N) + trailingZero(M);
}
 
// Driver Code
let N = 67;
let M = 98;
document.write(countProductTrailing(N, M));
 
// This code is contributed by _saurabh_jaiswal.
</script>

Producción: 
 

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