Maximum GCD of N integers with given product

Given N integers with unknown values (ai > 0) having product P. The task is to find the maximum possible greatest common divisor of these N integers.

Examples:

Input : N = 3, P = 24
Output : 2
The integers will have maximum GCD of 2 when a1 = 2, a2 = 2, a3 = 6.

Input : N = 2, P = 1
Output : 1
Only possibility is a1 = 1 and a2 = 1.

Approach:

  • First find all the prime factors of product P and store it in a Hashmap.
  • The N integers will have maximum GCD when a prime factor will be common in all the integers.
  • So if P = p1k1 * p2k2 * p3k3 …. where p1, p2 … are prime numbers then, maximum GCD which can be obtained will be ans = p1k1 / N * p2k2 / N * p3k3 / N ….

Below is the implementation of the above approach:

C++

filter_none

edit
close

play_arrow

link
brightness_4
code

// C++ implementation of above approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to find maximum GCD
// of N integers with product P
int maxGCD(int N, int P)
{
  
    int ans = 1;
  
    // map to store prime factors of P
    unordered_map<int, int> prime_factors;
  
    // prime factorization of P
    for (int i = 2; i * i <= P; i++) {
  
        while (P % i == 0) {
  
            prime_factors[i]++;
  
            P /= i;
        }
    }
  
    if (P != 1)
        prime_factors[P]++;
  
    // traverse all prime factors and
    // multiply its 1/N power to the result
    for (auto v : prime_factors) 
        ans *= pow(v.first, v.second / N);    
  
    return ans;
}
  
// Driver code
int main()
{
    int N = 3, P = 24;
  
    cout << maxGCD(N, P);
  
    return 0;
}

chevron_right


Java

filter_none

edit
close

play_arrow

link
brightness_4
code

// Java implementation of above approach 
import java.util.*;
class Solution
{
// Function to find maximum GCD 
// of N integers with product P 
static int maxGCD(int N, int P) 
  
    int ans = 1
  
    // map to store prime factors of P 
    Map<Integer, Integer> prime_factors =  
                        new HashMap< Integer,Integer>(); 
  
    // prime factorization of P 
    for (int i = 2; i * i <= P; i++) { 
  
        while (P % i == 0) { 
  
            if(prime_factors.get(i)==null)
            prime_factors.put(i,1);
            else
            prime_factors.put(i,(prime_factors.get(i)+1));
              
  
            P /= i; 
        
    
  
    if (P != 1
            if(prime_factors.get(P)==null)
            prime_factors.put(P,1);
            else
            prime_factors.put(P,(prime_factors.get(P)+1)); 
  
    // traverse all prime factors and 
    // multiply its 1/N power to the result 
        Set< Map.Entry< Integer,Integer> > st = prime_factors.entrySet();    
    
       for (Map.Entry< Integer,Integer> me:st) 
       
             
        ans *= Math.pow(me.getKey(),me.getValue() / N);    
        }
  
    return ans; 
  
// Driver code 
public static void main(String args[])
    int N = 3, P = 24
  
    System.out.println( maxGCD(N, P)); 
  
}
//contributed by Arnab Kundu

chevron_right


Python3

filter_none

edit
close

play_arrow

link
brightness_4
code

# Python3 implementation of 
# above approach 
from math import sqrt
  
# Function to find maximum GCD 
# of N integers with product P 
def maxGCD(N, P):
  
    ans = 1
  
    # map to store prime factors of P 
    prime_factors = {}
      
    # prime factorization of P 
    for i in range(2, int(sqrt(P) + 1)) :
  
        while (P % i == 0) :
              
            if i not in prime_factors :
                prime_factors[i] = 0
          
            prime_factors[i] += 1
            P //= i
          
    if (P != 1) :
        prime_factors[P] += 1
  
    # traverse all prime factors and 
    # multiply its 1/N power to the result 
    for key, value in prime_factors.items() :
        ans *= pow(key, value // N) 
  
    return ans
  
# Driver code 
if __name__ == "__main__"
  
    N, P = 3, 24
  
    print(maxGCD(N, P))
  
# This code is contributed by Ryuga

chevron_right


Output:

2


My Personal Notes arrow_drop_up

Check out this Author's contributed articles.

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 Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.



Improved By : andrew1234, AnkitRai01