Largest Subset with GCD 1

Last Updated : 13 Feb, 2025

Given n integers, we need to find size of the largest subset with GCD equal to 1.
Input Constraint : n <= 10^5, A[i] <= 10^5

Examples:

Input : A = {2, 3, 5}
Output : 3
Explanation: The largest subset with a GCD greater than 1 is {2, 3, 5}, and the GCD of all the elements in the subset is 3.

Input : A = {3, 18, 12}
Output : -1
Explanation: No subset of the array has a GCD greater than 1, so the output is -1.

[Naive Approach] by Evaluating All Subsets - O(n * log(n) * 2^n) Time and O(n) Space

We find GCD of all possible subsets and find the largest subset whose GCD is 1. Total time taken will be equal to the time taken to evaluate GCD of all possible subsets. Total possible subsets are 2ⁿ. In worst case there are n elements in subset and time taken to calculate its GCD will be n * log(n)
Extra space required to hold current subset is O(n)

[Expected Approach] By Checking GCD of Complete Set - O(n* log(n)) Time and O(1) Space

Let say we find a subset with GCD 1, if we add a new element to it then GCD still remains 1. Hence if a subset exists with GCD 1 then GCD of the complete set is also 1. Hence we first find GCD of the complete set, if its 1 then complete set is that subset else no subset exist with GCD 1.

C++

// C++ program to find size of the largest subset with GCD 1 #include <iostream> using namespace std;  // Function to return gcd of a and b int gcd(int a, int b) {     if (a == 0)         return b;     return gcd(b%a, a); }  // Function to find largest subset with GCD 1 int largestGCD1Subset(int A[], int n) {     // finding gcd of whole array     int currentGCD = A[0];     for (int i=1; i<n; i++)     {         currentGCD = gcd(currentGCD, A[i]);          // If current GCD becomes 1 at any moment,         // then whole array has GCD 1.         if (currentGCD == 1)             return n;     }      return 0; }  // Driver program to test above function int main() {     int A[] = {2, 18, 6, 3};     int n = sizeof(A)/sizeof(A[0]);     cout << largestGCD1Subset(A, n);     return 0; }

Java

// Java program to find size of the // largest subset with GCD 1 import java.io.*;  class GFG {          // Function to return gcd of     // a and b     static int gcd(int a, int b)     {         if (a == 0)             return b;         return gcd(b % a, a);     }      // Function to find largest     // subset with GCD 1     static int largestGCD1Subset(int A[],                                    int n)     {                  // finding gcd of whole array         int currentGCD = A[0];                  for (int i=1; i<n; i++)         {             currentGCD =                      gcd(currentGCD, A[i]);                  // If current GCD becomes 1             // at any moment, then whole             // array has GCD 1.             if (currentGCD == 1)                 return n;         }              return 0;     }          // Driver code     public static void main (String[] args)     {         int A[] = {2, 18, 6, 3};         int n =A.length;                  System.out.println(                   largestGCD1Subset(A, n) );     } }  // This code is contributed by Sam007.

Python

# python program to find size of the # largest subset with GCD 1  # Function to return gcd of a and b def gcd( a, b):          if (a == 0):         return b              return gcd(b%a, a)   # Function to find largest subset # with GCD 1 def largestGCD1Subset(A, n):          # finding gcd of whole array     currentGCD = A[0];     for i in range(1, n):                  currentGCD = gcd(currentGCD, A[i])          # If current GCD becomes 1 at         # any moment, then whole          # array has GCD 1.         if (currentGCD == 1):             return n     return 0  # Driver code A = [2, 18, 6, 3] n = len(A) print (largestGCD1Subset(A, n))  # This code is Contributed by Sam007.

// C# program to find size of the // largest subset with GCD 1 using System;          public class GFG {          // Function to return gcd of     // a and b     static int gcd(int a, int b)     {         if (a == 0)             return b;         return gcd(b % a, a);     }      // Function to find largest subset     // with GCD 1     static int largestGCD1Subset(int []A,                                    int n)     {                  // finding gcd of whole array         int currentGCD = A[0];         for (int i = 1; i < n; i++)         {             currentGCD =                      gcd(currentGCD, A[i]);                  // If current GCD becomes 1 at             // any moment, then whole             // array has GCD 1.             if (currentGCD == 1)                 return n;         }              return 0;     }          // Driver method      public static void Main()     {         int []A = {2, 18, 6, 3};         int n = A.Length;                  Console.Write(               largestGCD1Subset(A, n));     } }  // This code is contributed by Sam007.

JavaScript

<script>  // Javascript program to find size of the // largest subset with GCD 1  // Function to return gcd of a and b function gcd(a, b) {     if (a == 0)         return b;     return gcd(b % a, a); }  // Function to find largest subset // with GCD 1 function largestGCD1Subset(A, n) {          // finding gcd of whole array     let currentGCD = A[0];     for ( let i = 1; i < n; i++)     {         currentGCD =             gcd(currentGCD, A[i]);          // If current GCD becomes 1          // at any moment, then          // whole array has GCD 1.         if (currentGCD == 1)             return n;     }      return 0; }  // Driver program let A = [2, 18, 6, 3]; let n = A.length; document.write(largestGCD1Subset(A, n));      // This code is contributed by _saurabh_jaiswal  </script>

PHP

<?php  // php program to find size of the // largest subset with GCD 1  // Function to return gcd of a and b function gcd($a, $b) {     if ($a == 0)         return $b;     return gcd($b % $a, $a); }  // Function to find largest subset // with GCD 1 function largestGCD1Subset($A, $n) {          // finding gcd of whole array     $currentGCD = $A[0];     for ( $i = 1; $i < $n; $i++)     {         $currentGCD =             gcd($currentGCD, $A[$i]);          // If current GCD becomes 1          // at any moment, then          // whole array has GCD 1.         if ($currentGCD == 1)             return $n;     }      return 0; }  // Driver program $A = array(2, 18, 6, 3); $n = sizeof($A); echo largestGCD1Subset($A, $n);      // This code is contributed by ajit  ?>

Output

Time Complexity : O(n* log(n))
Space Complexity : O(1)

Summation of GCD of all the pairs up to n

Pratik Chhajer

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Largest Subset with GCD 1

[Naive Approach] by Evaluating All Subsets - O(n * log(n) * 2^n) Time and O(n) Space

[Expected Approach] By Checking GCD of Complete Set - O(n* log(n)) Time and O(1) Space

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