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.

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