Check if a number is divisible by all prime divisors of another number

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 the 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 divisors if and only if y/GCD has such unique prime divisor. So we have to find uniqueness for pair (x, y/GCD) using recursion.

Below is the implementation of the above approach:

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;

    int z = __gcd(x, y);

    if (z == 1)
        return false;

    return isDivisible(x, y / z);
}

// 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.
public 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
?>

Javascript

<script>

// Javascript 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;

    var z = gcd(x, y);

    if (z == 1)
        return false;

    return isDivisible(x, y / z);
}

// Driver program to test above functions

var x = 18, y = 12;
if (isDivisible(x, y))
    document.write("Yes");
else
    document.write("No");

// This code is contributed by Amit Katiyar

</script>

Output

Yes

Time Complexity: The 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(log²y)
Auxiliary Space: O(log min(x, y))

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