Given two integers. We need to find if the first number x is divisible by all prime divisors of y.

**Examples :**

Input : x = 120, y = 75 Output : Yes Explanation : 120 = (2^3)*3*5 75 = 3*(5^2) 120 is divisible by both 3 and 5 which are the prime divisors of 75. Hence, answer is "Yes". Input : x = 15, y = 6 Output : No Explanation : 15 = 3*5. 6 = 2*3, 15 is not divisible by 2 which is a prime divisor of 6. Hence, answer is "No".

A **simple solution** is to find all prime factors of y. For every prime factor, check if it divides x or not.

An **efficient solution** is based on below facts.

1) if y == 1, then it no prime divisors. Hence answer is “Yes”

2) We find GCD of x and y.

a) If GCD == 1, then clearly there are no common divisors of x and y, hence answer is “No”.

b) If GCD > 1, the GCD contains prime divisors which divide x also. Now, we have all unique prime divisor if and only if y/GCD has such unique prime divisor. So we have to find uniqueness for pair (x, y/GCD) using recursion.

## C++

`// CPP program to find if all prime factors` `// of y divide x.` `#include <bits/stdc++.h>` `using` `namespace` `std;` ` ` `// Returns true if all prime factors of y ` `// divide x.` `bool` `isDivisible(` `int` `x, ` `int` `y)` `{` ` ` `if` `(y == 1)` ` ` `return` `true` `;` ` ` ` ` `if` `(__gcd(x, y) == 1)` ` ` `return` `false` `;` ` ` `return` `isDivisible(x, y / gcd);` `}` ` ` `// Driver Code` `int` `main()` `{` ` ` `int` `x = 18, y = 12;` ` ` `if` `(isDivisible(x, y))` ` ` `cout << ` `"Yes"` `<< endl;` ` ` `else` ` ` `cout << ` `"No"` `<< endl;` ` ` `return` `0;` `}` |

## Java

`// Java program to find if all ` `// prime factors of y divide x.` `class` `Divisible ` `{` ` ` `public` `static` `int` `gcd(` `int` `a, ` `int` `b) { ` ` ` `return` `b == ` `0` `? a : gcd(b, a % b); }` ` ` ` ` `// Returns true if all prime factors` ` ` `// of y divide x.` ` ` `static` `boolean` `isDivisible(` `int` `x, ` `int` `y)` ` ` `{` ` ` `if` `(y == ` `1` `)` ` ` `return` `true` `;` ` ` ` ` `int` `z = gcd(x, y);` ` ` ` ` `if` `(z == ` `1` `)` ` ` `return` `false` `;` ` ` ` ` `return` `isDivisible(x, y / z);` ` ` `}` ` ` ` ` `// Driver program to test above functions` ` ` `public` `static` `void` `main(String[] args) ` ` ` `{` ` ` `int` `x = ` `18` `, y = ` `12` `;` ` ` `if` `(isDivisible(x, y))` ` ` `System.out.println(` `"Yes"` `);` ` ` `else` ` ` `System.out.println(` `"No"` `);` ` ` `}` `}` `// This code is contributed by Prerna Saini` |

## Python3

`# python program to find if all` `# prime factors of y divide x.` ` ` `def` `gcd(a, b):` ` ` `if` `(b ` `=` `=` `0` `):` ` ` `return` `a ` ` ` `else` `:` ` ` `return` `gcd(b, a ` `%` `b)` ` ` `# Returns true if all prime ` `# factors of y divide x.` `def` `isDivisible(x,y):` ` ` ` ` `if` `(y ` `=` `=` `1` `):` ` ` `return` `1` ` ` ` ` `z ` `=` `gcd(x, y);` ` ` ` ` `if` `(z ` `=` `=` `1` `):` ` ` `return` `false;` ` ` ` ` `return` `isDivisible(x, y ` `/` `z);` ` ` `# Driver Code` `x ` `=` `18` `y ` `=` `12` `if` `(isDivisible(x, y)):` ` ` `print` `(` `"Yes"` `)` `else` `:` ` ` `print` `(` `"No"` `)` ` ` `# This code is contributed by Sam007` |

## C#

`// C# program to find if all ` `// prime factors of y divide x.` `using` `System;` ` ` `class` `GFG {` ` ` ` ` `public` `static` `int` `gcd(` `int` `a, ` `int` `b)` ` ` `{ ` ` ` `return` `b == 0 ? a : gcd(b, a % b);` ` ` `}` ` ` ` ` `// Returns true if all prime factors` ` ` `// of y divide x.` ` ` `static` `bool` `isDivisible(` `int` `x, ` `int` `y)` ` ` `{` ` ` `if` `(y == 1)` ` ` `return` `true` `;` ` ` ` ` `int` `z = gcd(x, y);` ` ` ` ` `if` `(z == 1)` ` ` `return` `false` `;` ` ` ` ` `return` `isDivisible(x, y / z);` ` ` `}` ` ` ` ` `// Driver program to test above functions` ` ` `public` `static` `void` `Main() ` ` ` `{` ` ` `int` `x = 18, y = 12;` ` ` ` ` `if` `(isDivisible(x, y))` ` ` `Console.WriteLine(` `"Yes"` `);` ` ` `else` ` ` `Console.WriteLine(` `"No"` `);` ` ` `}` `}` ` ` `// This code is contributed by vt_m.` |

## PHP

`<?php` `// PHP program to find if all ` `// prime factors of y divide x.` ` ` `function` `gcd (` `$a` `, ` `$b` `) ` `{` ` ` `return` `$b` `== 0 ? ` `$a` `: ` ` ` `gcd(` `$b` `, ` `$a` `% ` `$b` `);` `} ` ` ` `// Returns true if all prime ` `// factors of y divide x.` `function` `isDivisible(` `$x` `, ` `$y` `)` `{` ` ` `if` `(` `$y` `== 1)` ` ` `return` `true;` ` ` ` ` `$z` `= gcd(` `$x` `, ` `$y` `);` ` ` ` ` `if` `(` `$z` `== 1)` ` ` `return` `false;` ` ` ` ` `return` `isDivisible(` `$x` `, ` `$y` `/ ` `$z` `);` `}` ` ` `// Driver Code` `$x` `= 18;` `$y` `= 12;` `if` `(isDivisible(` `$x` `, ` `$y` `))` ` ` `echo` `"Yes"` `;` `else` ` ` `echo` `"No"` `;` ` ` `// This code is contributed by Sam007` `?>` |

**Output :**

Yes

**Time Complexity:**Time complexity for calculating GCD is O(log min(x, y)), and recursion will terminate after log y steps because we are reducing it by a factor greater than one. Overall Time complexity: **O(log ^{2}y)**

This article is contributed by **Harsha Mogali**. 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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