Find the first natural number whose factorial is divisible by x

Given a number x, the task is to find first natural number i whose factorial is divisible by x.

Examples :

Input  : x = 10
Output : 5
5 is the smallest number such that 
(5!) % 10 = 0

Input  : x = 16
Output : 6
6 is the smallest number such that 
(6!) % 16 = 0

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

A simple solution is to iterate from 1 to x-1 and for every number i check if i! is divisible by x.

C++

// A simple C++ program to find first natural 
// number whose factorial divides x. 
#include <bits/stdc++.h> 
using namespace std; 
  
// Returns first number whose factorial 
// divides x. 
int firstFactorialDivisibleNumber(int x) 
{ 
    int i = 1; // Result 
    int fact = 1; 
    for (i = 1; i < x; i++) { 
        fact = fact * i; 
        if (fact % x == 0) 
            break; 
    } 
  
    return i; 
} 
  
// Driver code 
int main(void) 
{ 
    int x = 16; 
    cout << firstFactorialDivisibleNumber(x); 
    return 0; 
} 

Java

// A simple Java program to find first natural 
// number whose factorial divides x 
class GFG { 
  
    // Returns first number whose factorial 
    // divides x. 
    static int firstFactorialDivisibleNumber(int x) 
    { 
        int i = 1; // Result 
        int fact = 1; 
        for (i = 1; i < x; i++) { 
            fact = fact * i; 
            if (fact % x == 0) 
                break; 
        } 
  
        return i; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        int x = 16; 
        System.out.print(firstFactorialDivisibleNumber(x)); 
    } 
} 
  
// This code is contributed by Anant Agarwal. 

Python3

# A simple python program to find  
# first natural number whose  
# factorial divides x. 
  
# Returns first number whose  
# factorial divides x. 
def firstFactorialDivisibleNumber(x): 
    i = 1; # Result 
    fact = 1; 
    for i in range(1, x): 
        fact = fact * i 
        if (fact % x == 0): 
            break
    return i 
  
# Driver code 
x = 16
print(firstFactorialDivisibleNumber(x)) 
  
# This code is contributed  
# by 29AjayKumar 

C#

// A simple C# program to find first natural 
// number whose factorial divides x 
using System; 
  
class GFG { 
  
    // Returns first number whose factorial 
    // divides x. 
    static int firstFactorialDivisibleNumber(int x) 
    { 
        int i = 1; // Result 
        int fact = 1; 
        for (i = 1; i < x; i++) { 
            fact = fact * i; 
            if (fact % x == 0) 
                break; 
        } 
  
        return i; 
    } 
  
    // Driver code 
    public static void Main() 
    { 
        int x = 16; 
  
        Console.Write( 
            firstFactorialDivisibleNumber(x)); 
    } 
} 
  
// This code is contributed by nitin mittal 

PHP

<?php 
// A simple PHP program to find 
// first natural number whose  
// factorial divides x. 
  
// Returns first number whose 
// factorial divides x. 
function firstFactorialDivisibleNumber($x) 
{ 
    // Result 
    $i = 1;  
    $fact = 1; 
    for ($i = 1; $i < $x; $i++)  
    { 
        $fact = $fact * $i; 
        if ($fact % $x == 0) 
            break; 
    } 
  
    return $i; 
} 
  
// Driver code 
$x = 16; 
echo(firstFactorialDivisibleNumber($x)); 
  
// This code is contributed by Ajit. 
?> 

Output :

If we apply this naive approach, we wouldn’t go above 20! or 21! (long long int will have its upper limit).

A better solution avoids overflow. The solution is based on below observations.

If i! is divisible by x, then (i+1)!, (i+2)!, … are also divisible by x.
For a number x, all factorials i! are divisible by x when i >= x.
If a number x is prime, then no factorial below x can divide it as x cannot be formed with multiplication of smaller numbers.

Below is algorithm

1) Run a loop for i = 1 to n-1
       
   a) Remove common factors
      new_x /= gcd(i, new_x);

   b) Check if we found first i.
      if (new_x == 1)
          break;

2) Return i

Below is the implementation of above idea :

CPP

// C++ program to find first natural number 
// whose factorial divides x. 
#include <bits/stdc++.h> 
using namespace std; 
  
// GCD function to compute the greatest 
// divisor among a and b 
int gcd(int a, int b) 
{ 
    if ((a % b) == 0) 
        return b; 
    return gcd(b, a % b); 
} 
  
// Returns first number whose factorial 
// divides x. 
int firstFactorialDivisibleNumber(int x) 
{ 
    int i = 1; // Result 
    int new_x = x; 
  
    for (i = 1; i < x; i++) { 
        // Remove common factors 
        new_x /= gcd(i, new_x); 
  
        // We found first i. 
        if (new_x == 1) 
            break; 
    } 
    return i; 
} 
  
// Driver code 
int main(void) 
{ 
    int x = 16; 
    cout << firstFactorialDivisibleNumber(x); 
    return 0; 
} 

Java

// Efficient Java program to find first 
// natural number whose factorial divides x. 
class GFG { 
  
    // GCD function to compute the greatest 
    // divisor among a and b 
    static int gcd(int a, int b) 
    { 
        if ((a % b) == 0) 
            return b; 
        return gcd(b, a % b); 
    } 
  
    // Returns first number whose factorial 
    // divides x. 
    static int firstFactorialDivisibleNumber(int x) 
    { 
        int i = 1; // Result 
        int new_x = x; 
  
        for (i = 1; i < x; i++) { 
  
            // Remove common factors 
            new_x /= gcd(i, new_x); 
  
            // We found first i. 
            if (new_x == 1) 
                break; 
        } 
        return i; 
    } 
  
    // Driver code 
    public static void main(String[] args) 
    { 
        int x = 16; 
        System.out.print(firstFactorialDivisibleNumber(x)); 
    } 
} 
// This code is contributed by Anant Agarwal. 

Python3

      
#  Python3 program to find first natural number 
#  whose factorial divides x. 
  
   
#  GCD function to compute the greatest 
#  divisor among a and b 
def gcd(a,  b): 
    if ((a % b) == 0): 
        return b 
    return gcd(b, a % b) 
  
   
#  Returns first number whose factorial 
#  divides x. 
def firstFactorialDivisibleNumber(x): 
    i = 1 #  Result 
    new_x = x 
   
    for i in range(1,x): 
        #  Remove common factors 
        new_x /= gcd(i, new_x) 
   
        #  We found first i. 
        if (new_x == 1): 
            break
    return i 
   
#  Driver code 
def main(): 
    x = 16
    print(firstFactorialDivisibleNumber(x)) 
  
if __name__ == '__main__': 
    main() 
  
# This code is contributed by 29AjayKumar  

C#

// Efficient C# program to find first 
// natural number whose factorial  
// divides x. 
using System; 
  
class GFG { 
  
    // GCD function to compute the 
    // greatest divisor among a 
    // and b 
    static int gcd(int a, int b) 
    { 
        if ((a % b) == 0) 
            return b; 
        return gcd(b, a % b); 
    } 
  
    // Returns first number whose 
    // factorial divides x. 
    static int firstFactorialDivisibleNumber( 
                                        int x) 
    { 
        int i = 1; // Result 
        int new_x = x; 
  
        for (i = 1; i < x; i++) { 
  
            // Remove common factors 
            new_x /= gcd(i, new_x); 
  
            // We found first i. 
            if (new_x == 1) 
                break; 
        } 
          
        return i; 
    } 
  
    // Driver code 
    public static void Main() 
    { 
        int x = 16; 
        Console.Write( 
            firstFactorialDivisibleNumber(x)); 
    } 
} 
  
// This code is contributed by nitin mittal. 

PHP

<?php 
// PHP program to find first  
// natural number whose  
// factorial divides x. 
  
// GCD function to compute the  
// greatest divisor among a and b 
function gcd($a, $b) 
{ 
    if (($a % $b) == 0) 
        return $b; 
    return gcd($b, $a % $b); 
} 
  
// Returns first number  
// whose factorial divides x. 
function firstFactorialDivisibleNumber($x) 
{ 
    // Result 
    $i = 1;  
    $new_x = $x; 
  
    for ($i = 1; $i < $x; $i++)  
    { 
        // Remove common factors 
        $new_x /= gcd($i, $new_x); 
  
        // We found first i. 
        if ($new_x == 1) 
            break; 
    } 
    return $i; 
} 
  
// Driver code 
$x = 16; 
echo(firstFactorialDivisibleNumber($x)); 
  
// This code is contributed by Ajit. 
?> 

Output :

Another approach using boost library:
(Thanking ajay0007 for contributing this approach)
Here we use boost library to efficiently calculate the value of factorial.
Prerequisite :boost-multiprecision-library

C++

// A cpp program for finding  
// the Special Factorial Number 
#include <bits/stdc++.h> 
#include <boost/multiprecision/cpp_int.hpp> 
  
using boost::multiprecision::cpp_int; 
using namespace std; 
  
// function for calculating factoial 
cpp_int fact(int n) 
{ 
    cpp_int num = 1; 
      
    for (int i = 1; i <= n; i++) 
        num = num * i; 
      
    return num; 
} 
  
// function for check Special_Factorial_Number 
int Special_Factorial_Number(int k) 
{ 
      
    for(int i = 1 ; i <= k ; i++ ) 
    {  
        // call fact function and the  
        // Modulo with k and check 
        // if condition is TRUE then return i 
        if ( ( fact (i) % k ) == 0 ) 
        { 
            return i; 
        } 
    } 
} 
  
//driver function 
int main() 
{ 
    // taking input 
    int k = 16; 
      
    cout<<Special_Factorial_Number(k); 
} 

Java

// Java program for finding  
// the Special Factorial Number  
public class GFG { 
  
// function for calculating factoial  
    static int fact(int n) { 
        int num = 1; 
  
        for (int i = 1; i <= n; i++) { 
            num = num * i; 
        } 
  
        return num; 
    } 
  
// function for check Special_Factorial_Number  
    static int Special_Factorial_Number(int k) { 
  
        for (int i = 1; i <= k; i++) { 
            // call fact function and the  
            // Modulo with k and check  
            // if condition is TRUE then return i  
            if (fact(i) % k == 0) { 
                return i; 
            } 
        } 
        return 0; 
    } 
  
//driver function  
    public static void main(String[] args) { 
        // taking input  
        int k = 16; 
        System.out.println(Special_Factorial_Number(k)); 
  
    } 
} 
  
/*This code is contributed by Rajput-Ji*/

Python3

# Python 3 program for finding  
# the Special Factorial Number  
  
# function for calculating factoial  
def fact( n): 
    num = 1
    for i in range(1, n + 1): 
        num = num * i 
    return num 
  
# function for check Special_Factorial_Number  
def Special_Factorial_Number(k): 
      
    for i in range(1, k + 1): 
          
        # call fact function and the  
        # Modulo with k and check  
        # if condition is TRUE then return i  
        if (fact(i) % k == 0): 
            return i 
    return 0
  
# Driver Code 
if __name__ == '__main__': 
      
    # taking input  
    k = 16
    print(Special_Factorial_Number(k)) 
  
# This code is contributed by Rajput-Ji 

C#

// C# program for finding  
// the Special Factorial Number  
using System;  
public class GFG{ 
  
  
// function for calculating factoial  
    static int fact(int n) {  
        int num = 1;  
  
        for (int i = 1; i <= n; i++) {  
            num = num * i;  
        }  
  
        return num;  
    }  
  
// function for check Special_Factorial_Number  
    static int Special_Factorial_Number(int k) {  
  
        for (int i = 1; i <= k; i++) {  
            // call fact function and the  
            // Modulo with k and check  
            // if condition is TRUE then return i  
            if (fact(i) % k == 0) {  
                return i;  
            }  
        }  
        return 0;  
    }  
  
//driver function  
    public static void Main() {  
        // taking input  
        int k = 16;  
        Console.WriteLine(Special_Factorial_Number(k));  
  
    }  
}  
  
// This code is contributed by 29AjayKumar 

PHP

<?php 
// PHP program for finding  
// the Special Factorial Number 
  
// function for calculating  
// factoial 
function fact($n) 
{ 
    $num = 1; 
      
    for ($i = 1; $i <= $n; $i++) 
        $num = $num * $i; 
      
    return $num; 
} 
  
// function for check  
// Special_Factorial_Number 
function Special_Factorial_Number($k) 
{ 
      
    for($i = 1 ; $i <= $k ; $i++ ) 
    {  
          
        // call fact function and the  
        // Modulo with k and check 
        // if condition is TRUE  
        // then return i 
        if (( fact ($i) % $k ) == 0 ) 
        { 
            return $i; 
        } 
    } 
} 
  
    // Driver Code 
    $k = 16; 
    echo Special_Factorial_Number($k); 
  
// This code is contributed by Ajit. 
?> 

Output :

This article is contributed by Shubham Gupta. 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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

My Personal Notes

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

C++

Java

Python3

C#

PHP

CPP

Java

Python3

C#

PHP

C++

Java

Python3

C#

PHP

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