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# 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```

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 ``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

 ``

## Javascript

 ``

Output

`6`

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 the above idea :

## C++

 `// C++ program to find first natural number``// whose factorial divides x.``#include ``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

 ``

## Javascript

 ``

Output

`6`

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 ``#include ` `using` `boost::multiprecision::cpp_int;``using` `namespace` `std;` `// function for calculating factorial``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<

## Java

 `// Java program for finding``// the Special Factorial Number``public` `class` `GFG {` `// function for calculating factorial``    ``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 factorial``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 factorial``    ``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

 ``

## Javascript

 ``

Output :

`6`

Time complexity: O(n^2) since using a for loop in another for loop

Auxiliary Space: O(1) because it is using constant variable

### Method 4: Using a generator

This code uses generators and loops to find the first natural number whose factorial is divisible by a given number x.

The factorial_gen() function is a generator that yields the factorial of each natural number starting from 1. It uses a while loop that keeps multiplying the current factorial with the next natural number and yields the result at each iteration.

The factorial_divisible_by_x() function takes the input x and loops through the factorials generated by factorial_gen(). It checks if the current factorial is divisible by x using the modulo operator. If it is, then the function returns the index of the natural number (i+1) whose factorial was found to be divisible by x.

Finally, the code calls the factorial_divisible_by_x() function with x = 10 and prints the result.

## C++

 `#include ``using` `namespace` `std;` `int` `main()``{``    ``int` `fact = 1;``    ``int` `i = 1;``    ``int` `x = 10;``    ``int` `count = 0;` `    ``while` `(``true``) {``        ``if` `(fact % x == 0) {``            ``count = i;``            ``break``;``        ``}``        ``i++;``        ``fact *= i;``    ``}` `    ``cout << count << endl;` `    ``return` `0;``}`

## Java

 `public` `class` `Main {``    ``public` `static` `void` `main(String[] args)``    ``{``        ``int` `fact = ``1``; ``// initialize factorial to 1``        ``int` `i = ``1``; ``// initialize counter to 1``        ``int` `x = ``10``; ``// set the value of x``        ``int` `count = ``0``; ``// initialize count to 0``        ``while` `(``true``) { ``// loop until a break statement is``                       ``// executed``            ``if` `(fact % x == ``0``) { ``// check if the factorial``                                 ``// is divisible by x``                ``count = i; ``// set the count to the current``                           ``// value of i``                ``break``; ``// exit the loop``            ``}``            ``i++; ``// increment the counter``            ``fact *= i; ``// calculate the factorial``        ``}``        ``System.out.println(count); ``// print the count``    ``}``}`

## Python3

 `def` `factorial_gen():``    ``fact ``=` `1``    ``i ``=` `1``    ``while` `True``:``        ``yield` `fact``        ``i ``+``=` `1``        ``fact ``*``=` `i`  `def` `factorial_divisible_by_x(x):``    ``for` `i, f ``in` `enumerate``(factorial_gen()):``        ``if` `f ``%` `x ``=``=` `0``:``            ``return` `i``+``1`  `x ``=` `10``print``(factorial_divisible_by_x(x))`

## Javascript

 `// Example usage``let x = 10;``let fact = 1;``let i = 1;``let count = 0;` `while` `(``true``) {``    ``if` `(fact % x === 0) {``        ``count = i;``        ``break``;``    ``}``    ``i++;``    ``fact *= i;``}` `console.log(count);`

## C#

 `using` `System;` `class` `Program {``    ``static` `void` `Main(``string``[] args) {``        ``int` `fact = 1;``        ``int` `i = 1;``        ``int` `x = 10;``        ``int` `count = 0;` `        ``while` `(``true``) {``            ``if` `(fact % x == 0) {``                ``count = i;``                ``break``;``            ``}``            ``i++;``            ``fact *= i;``        ``}` `        ``Console.WriteLine(count);``    ``}``}`

Output

`5`

The time complexity of the factorial_divisible_by_x function depends on the value of x and the number of iterations it takes to find the first factorial that is divisible by x. The factorial_gen generator function has a time complexity of O(n) as it generates factorials endlessly.

The space complexity of both functions is O(1) as they only use a fixed number of variables to store the factorial and the index. The factorial_gen generator does not store all the factorials it generates at once, but only the latest one.

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