# Largest Subset with GCD 1

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 Input : A = {3, 18, 12} Output : -1

## Naive Solution :

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 2^{n}. 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)

Time complexity : O(n * log(n) * 2^n) Space Complexity : O(n)

## Optimized O(n) solution :

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.*;` `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.` |

## Python3

`# 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#

`// 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.` |

## 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 ` `?>` |

## 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>` |

**Output: **

4

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

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