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

  • Difficulty Level : Medium
  • Last Updated : 13 Apr, 2021

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".

 

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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
?>

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: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)
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.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 




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