# GCD of factorials of two numbers

Given two numbers m and n. Find the GCD of the their factorial.**Examples :**

Input : n = 3, m = 4 Output : 6 Explanation: Factorial of n = 1 * 2 * 3 = 6 Factorial of m = 1 * 2 * 3 * 4 = 24 GCD(6, 24) = 6. Input : n = 9, m = 5 Output : 20 Explanation: Factorial of n = 1 * 2 * 3 *4 * 5 * 6 * 7 * 8 * 9 = 362880 Factorial of m = 1 * 2 * 3 * 4 * 5 = 120 GCD(362880, 120) = 120

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A **simple solution** is to find factorials of both numbers. Then find GCD of the computed factorials.

An **efficient solution** is based on the fact that GCD of two factorials is equal to smaller factorial (note that factorials have all terms common).

Below is the implementation of above approach.

## C++

`// CPP program to find GCD of factorial of two` `// numbers.` `#include <bits/stdc++.h>` `using` `namespace` `std;` `int` `factorial(` `int` `x)` `{` ` ` `if` `(x <= 1)` ` ` `return` `1;` ` ` `int` `res = 2;` ` ` `for` `(` `int` `i = 3; i <= x; i++)` ` ` `res = res * i;` ` ` `return` `res;` `}` `int` `gcdOfFactorial(` `int` `m, ` `int` `n)` `{` ` ` `return` `factorial(min(m, n));` `}` `int` `main()` `{` ` ` `int` `m = 5, n = 9;` ` ` `cout << gcdOfFactorial(m, n);` ` ` `return` `0;` `}` |

## Java

`// Java program to find GCD of factorial` `// of two numbers.` `public` `class` `FactorialGCD{` ` ` `static` `int` `factorial(` `int` `x)` `{` ` ` `if` `(x <= ` `1` `)` ` ` `return` `1` `;` ` ` `int` `res = ` `2` `;` ` ` `for` `(` `int` `i = ` `3` `; i <= x; i++)` ` ` `res = res * i;` ` ` `return` `res;` `}` `static` `int` `gcdOfFactorial(` `int` `m, ` `int` `n)` `{` ` ` `int` `min = m < n ? m : n;` ` ` `return` `factorial(min);` `}` ` ` `/* Driver program to test above functions */` ` ` `public` `static` `void` `main (String[] args)` ` ` `{` ` ` `int` `m = ` `5` `, n = ` `9` `;` ` ` ` ` `System.out.println(gcdOfFactorial(m, n));` ` ` `}` `}` `// This code is contributed by Prerna Saini` |

## Python

`# Python code to find GCD of factorials of` `# two numbers.` `import` `math` `def` `gcdOfFactorial(m, n) :` ` ` `return` `math.factorial(` `min` `(m, n))` `# Driver code` `m ` `=` `5` `n ` `=` `9` `print` `(gcdOfFactorial(m, n))` |

## C#

`// C# program to find GCD of factorial` `// of two numbers.` `using` `System;` `public` `class` `GFG{` ` ` ` ` `static` `int` `factorial(` `int` `x)` ` ` `{` ` ` `if` `(x <= 1)` ` ` `return` `1;` ` ` ` ` `int` `res = 2;` ` ` ` ` `for` `(` `int` `i = 3; i <= x; i++)` ` ` `res = res * i;` ` ` ` ` `return` `res;` ` ` `}` ` ` ` ` `static` `int` `gcdOfFactorial(` `int` `m, ` `int` `n)` ` ` `{` ` ` `int` `min = m < n ? m : n;` ` ` `return` `factorial(min);` ` ` `}` ` ` ` ` `/* Driver program to test above functions */` ` ` `public` `static` `void` `Main()` ` ` `{` ` ` ` ` `int` `m = 5, n = 9;` ` ` ` ` `Console.WriteLine(gcdOfFactorial(m, n));` ` ` `}` `}` ` ` `// This code is contributed by Anant Agarwal.` |

## PHP

`<?php` `// PHP program to find GCD` `// of factorial of two numbers.` `function` `factorial(` `$x` `)` `{` ` ` `if` `(` `$x` `<= 1)` ` ` `return` `1;` ` ` `$res` `= 2;` ` ` `for` `(` `$i` `= 3; ` `$i` `<= ` `$x` `; ` `$i` `++)` ` ` `$res` `= ` `$res` `* ` `$i` `;` ` ` `return` `$res` `;` `}` `function` `gcdOfFactorial(` `$m` `, ` `$n` `)` `{` ` ` `return` `factorial(min(` `$m` `, ` `$n` `));` `}` `// Driver Code` `$m` `= 5; ` `$n` `= 9;` `echo` `gcdOfFactorial(` `$m` `, ` `$n` `);` `// This code is contributed by ajit` `?>` |

## Javascript

`<script>` `// JavaScript program to find GCD of factorial` `// of two numbers.` ` ` `function` `factorial(x) {` ` ` `if` `(x <= 1)` ` ` `return` `1;` ` ` `var` `res = 2;` ` ` `for` `(i = 3; i <= x; i++)` ` ` `res = res * i;` ` ` `return` `res;` ` ` `}` ` ` `function` `gcdOfFactorial(m , n) {` ` ` `var` `min = m < n ? m : n;` ` ` `return` `factorial(min);` ` ` `}` ` ` `/* Driver program to test above functions */` ` ` ` ` `var` `m = 5, n = 9;` ` ` `document.write(gcdOfFactorial(m, n));` `// This code is contributed by aashish1995` `</script>` |

**Output :**

120