Smallest number with at least n trailing zeroes in factorial

Given a number n. The task is to find the smallest number whose factorial contains at least n trailing zeroes.

Examples :

Input : n = 1
Output : 5 
1!, 2!, 3!, 4! does not contain trailing zero.
5! = 120, which contains one trailing zero.

Input : n = 6
Output : 25



In the article for Count trailing zeroes in factorial of a number, we have discussed number of zeroes is equal to number of 5’s in prime factors of x!. We have discussed below formula to count number of 5’s.

Trailing 0s in x! = Count of 5s in prime factors of x!
                  = floor(x/5) + floor(x/25) + floor(x/125) + ....

Let us take few examples to observe pattern 

5!  has 1 trailing zeroes 
[All numbers from 6 to 9
 have 1 trailing zero]

10! has 2 trailing zeroes
[All numbers from 11 to 14
 have 2 trailing zeroes]

15! to 19! have 3 trailing zeroes

20! to 24! have 4 trailing zeroes

25! to 29! have 6 trailing zeroes

We can notice that, the minimum value whose factorial contain n trailing zeroes is 5*n.

So, to find minimum value whose factorial contains n trailing zeroes, use binary search on range from 0 to 5*n. And, find the smallest number whose factorial contains n trailing zeroes.

C++

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// C++ program tofind smallest number whose
// factorial contains at least n trailing
// zeroes.
#include<bits/stdc++.h>
using namespace std;
  
// Return true if number's factorial contains
// at least n trailing zero else false.
bool check(int p, int n)
{
    int temp = p, count = 0, f = 5;
    while (f <= temp)
    {
        count += temp/f;
        f = f*5;
    }
    return (count >= n);
}
  
// Return smallest number whose factorial
// contains at least n trailing zeroes
int findNum(int n)
{
    // If n equal to 1, return 5.
    // since 5! = 120.
    if (n==1)
        return 5;
  
    // Initalising low and high for binary
    // search.
    int low = 0;
    int high = 5*n;
  
    // Binary Search.
    while (low <high)
    {
        int mid = (low + high) >> 1;
  
        // Checking if mid's factorial contains
        // n trailing zeroes.
        if (check(mid, n))
            high = mid;
        else
            low = mid+1;
    }
  
    return low;
}
  
// driver code
int main()
{
    int n = 6;
    cout << findNum(n) << endl;
    return 0;
}

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Java














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// Java program tofind smallest number whose 


// factorial contains at least n trailing 


// zeroes. 


  


class GFG 


{ 


    // Return true if number's factorial contains 


    // at least n trailing zero else false. 


    static boolean check(int p, int n) 


    { 


        int temp = p, count = 0, f = 5; 


        while (f <= temp) 


        { 


            count += temp / f; 


            f = f * 5; 


        } 


        return (count >= n); 


    }  


      


    // Return smallest number whose factorial 


    // contains at least n trailing zeroes 


    static int findNum(int n) 


    { 


        // If n equal to 1, return 5. 


        // since 5! = 120. 


        if (n==1) 


            return 5; 


      


        // Initalising low and high for binary 


        // search. 


        int low = 0; 


        int high = 5 * n; 


      


        // Binary Search. 


        while (low < high) 


        { 


            int mid = (low + high) >> 1; 


      


            // Checking if mid's factorial  


            // contains n trailing zeroes. 


            if (check(mid, n)) 


                high = mid; 


            else


                low = mid + 1; 


        } 


      


        return low; 


    } 


      


    // Driver code  


    public static void main (String[] args) 


    { 


        int n = 6; 


        System.out.println(findNum(n)); 


    } 


} 


  


// This code is contributed by Anant Agarwal. 



















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Python3














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# Python3 program tofind smallest 


# number whose 


# factorial contains at least 


# n trailing zeroes 


  


# Return true if number's factorial contains 


# at least n trailing zero else false. 


def check(p,n): 


  


    temp = p 


    count = 0


    f = 5


    while (f <= temp): 


        count += temp/f 


        f = f*5


  


    return (count >= n) 


  


# Return smallest number whose factorial 


# contains at least n trailing zeroes 


def findNum(n): 


  


    # If n equal to 1, return 5. 


    # since 5! = 120. 


    if (n==1): 


        return 5


   


    # Initalizing low and high for binary 


    # search. 


    low = 0


    high = 5*n 


   


    # Binary Search. 


    while (low <high): 


  


        mid = (low + high) >> 1


   


        # Checking if mid's factorial contains 


        # n trailing zeroes. 


        if (check(mid, n)): 


            high = mid 


        else: 


            low = mid+1


      


   


    return low 


  


  


# driver code 


n = 6


print(findNum(n)) 


  


# This code is contributed 


# by Anant Agarwal.  



















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C#














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// C# program tofind smallest number whose 


// factorial contains at least n trailing 


// zeroes. 


using System; 


  


class GFG 


{ 


    // Return true if number's factorial contains 


    // at least n trailing zero else false. 


    static bool check(int p, int n) 


    { 


        int temp = p, count = 0, f = 5; 


        while (f <= temp) 


        { 


            count += temp / f; 


            f = f * 5; 


        } 


        return (count >= n); 


    }  


      


    // Return smallest number whose factorial 


    // contains at least n trailing zeroes 


    static int findNum(int n) 


    { 


        // If n equal to 1, return 5. 


        // since 5! = 120. 


        if (n == 1) 


            return 5; 


      


        // Initalising low and high for binary 


        // search. 


        int low = 0; 


        int high = 5 * n; 


      


        // Binary Search. 


        while (low < high) 


        { 


            int mid = (low + high) >> 1; 


      


            // Checking if mid's factorial  


            // contains n trailing zeroes. 


            if (check(mid, n)) 


                high = mid; 


            else


                low = mid + 1; 


        } 


      


        return low; 


    } 


      


    // Driver code  


    public static void Main () 


    { 


        int n = 6; 


          


        Console.WriteLine(findNum(n)); 


    } 


} 


  


// This code is contributed by vt_m. 



















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PHP














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<?php 


// PHP program to find smallest 


// number whose factorial contains 


// at least n trailing zeroes. 


  


// Return true if number's  


// factorial contains at  


// least n trailing zero  


// else false. 


function check($p, $n) 


{ 


    $temp = $p; $count = 0; $f = 5; 


    while ($f <= $temp) 


    { 


        $count += $temp / $f; 


        $f = $f * 5; 


    } 


    return ($count >= $n); 


} 


  


// Return smallest number  


// whose factorial contains  


// at least n trailing zeroes 


function findNum($n) 


{ 


    // If n equal to 1, return 5. 


    // since 5! = 120. 


    if ($n == 1) 


        return 5; 


  


    // Initalising low and high 


    // for binary search. 


    $low = 0; 


    $high = 5 * $n; 


  


    // Binary Search. 


    while ($low < $high) 


    { 


        $mid = ($low + $high) >> 1; 


  


        // Checking if mid's factorial  


        // contains n trailing zeroes. 


        if (check($mid, $n)) 


            $high = $mid; 


        else


            $low = $mid + 1; 


    } 


  


    return $low; 


} 


  


// Driver Code 


$n = 6; 


echo(findNum($n)); 


  


// This code is contributed by Ajit. 


?> 



















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Output :


25



This article is contributed by Anuj Chauhan. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.


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