Smallest number with at least n trailing zeroes in factorial - GeeksforGeeks

08 Jun, 2021
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 maximum 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++
Java
Python3
C#
PHP
Javascript
// 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 zeroesint findNum(int n){    // If n equal to 1, return 5.    // since 5! = 120.    if (n==1)        return 5;     // Initialising 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 codeint main(){    int n = 6;    cout << findNum(n) << endl;    return 0;}
// 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;             // Initialising 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.
# 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 zeroesdef findNum(n):     # If n equal to 1, return 5.    # since 5! = 120.    if (n==1):        return 5      # Initializing 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 coden = 6print(findNum(n)) # This code is contributed# by Anant Agarwal.
// 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;             // Initialising 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.
<?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 >= $n);} // Return smallest number// whose factorial contains// at least n trailing zeroesfunction findNum($n){    // If n equal to 1, return 5.    // since 5! = 120.    if ($n == 1)        return 5;     // Initialising 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.?>
<script>// JavaScript 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){    let temp = p, count = 0, f = 5;    while (f <= temp)    {        count += Math.floor(temp/f);        f = f*5;    }    return (count >= n);} // Return smallest number whose factorial// contains at least n trailing zeroesfunction findNum(n){    // If n equal to 1, return 5.    // since 5! = 120.    if (n==1)        return 5;     // Initialising low and high for binary    // search.    let low = 0;    let high = 5*n;     // Binary Search.    while (low <high)    {        let 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    let n = 6;    document.write(findNum(n) + "<br>");  // This code is contributed by Surbhi Tyagi </script>
Output :  
25
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