# Check if all array elements are pairwise co-prime or not

Given an array A[] consisting of N positive integers, the task is to check if all the array elements are pairwise co-prime, i.e. for all pairs (Ai , Aj), such that 1<=i<j<=N, GCD(Ai, Aj) = 1.

Examples:

Input : A[] = {2, 3, 5}
Output : Yes
Explanation : All the pairs, (2, 3), (3, 5), (2, 5) are pairwise co-prime.

Input : A[] = {5, 10}
Output : No
Explanation : GCD(5, 10)=5 so they are not co-prime.

Naive Approach: The simplest approach to solve the problem is to generate all possible pairs from a given array and for each pair, check if it is coprime or not. If any pair is found to be non-coprime, print “No“. Otherwise, print “Yes“.
Time Complexity: O(N2)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the following observation:

If any two numbers have a common prime factor, then their GCD can never be 1.

This can also be interpreted as:

The LCM of the array must be equal to the product of the elements in the array.

Therefore, the solution boils down to calculating the LCM of the given array and check if it is equal to the product of all the array elements or not.

Below is the implementation of the above approach :

 `// C++ Program for the above approach ` `#include ` `using` `namespace` `std; ` `#define ll long long int `   `// Function to calculate GCD ` `ll GCD(ll a, ll b) ` `{ ` `    ``if` `(a == 0) ` `        ``return` `b; ` `    ``return` `GCD(b % a, a); ` `} `   `// Function to calculate LCM ` `ll LCM(ll a, ll b) ` `{ ` `    ``return` `(a * b) ` `        ``/ GCD(a, b); ` `} `   `// Function to check if all elements ` `// in the array are pairwise coprime ` `void` `checkPairwiseCoPrime(``int` `A[], ``int` `n) ` `{ ` `    ``// Initialze variables ` `    ``ll prod = 1; ` `    ``ll lcm = 1; `   `    ``// Itertae over the array ` `    ``for` `(``int` `i = 0; i < n; i++) { `   `        ``// Calculate product of ` `        ``// array elements ` `        ``prod *= A[i]; `   `        ``// Calculate LCM of ` `        ``// array elements ` `        ``lcm = LCM(A[i], lcm); ` `    ``} `   `    ``// If the product of array elements ` `    ``// is equal to LCM of the array ` `    ``if` `(prod == lcm) ` `        ``cout << ``"Yes"` `<< endl; ` `    ``else` `        ``cout << ``"No"` `<< endl; ` `} ` `// Driver Code ` `int` `main() ` `{ ` `    ``int` `A[] = { 2, 3, 5 }; ` `    ``int` `n = ``sizeof``(A) / ``sizeof``(A[0]); `   `    ``// Function call ` `    ``checkPairwiseCoPrime(A, n); ` `}`

 `// Java program for the above approach` `import` `java.util.*;` `import` `java.lang.*;`   `class` `GFG{`   `// Function to calculate GCD ` `static` `long` `GCD(``long` `a, ``long` `b) ` `{ ` `    ``if` `(a == ``0``) ` `        ``return` `b; ` `        `  `    ``return` `GCD(b % a, a); ` `} `   `// Function to calculate LCM ` `static` `long` `LCM(``long` `a, ``long` `b) ` `{ ` `    ``return` `(a * b) / GCD(a, b); ` `} `   `// Function to check if all elements ` `// in the array are pairwise coprime ` `static` `void` `checkPairwiseCoPrime(``int` `A[], ``int` `n) ` `{ ` `    `  `    ``// Initialze variables ` `    ``long` `prod = ``1``; ` `    ``long` `lcm = ``1``; `   `    ``// Itertae over the array ` `    ``for``(``int` `i = ``0``; i < n; i++)` `    ``{ ` `        `  `        ``// Calculate product of ` `        ``// array elements ` `        ``prod *= A[i]; `   `        ``// Calculate LCM of ` `        ``// array elements ` `        ``lcm = LCM(A[i], lcm); ` `    ``} ` `    `  `    ``// If the product of array elements ` `    ``// is equal to LCM of the array ` `    ``if` `(prod == lcm) ` `        ``System.out.println(``"Yes"``); ` `    ``else` `        ``System.out.println(``"No"``); ` `}`   `// Driver Code` `public` `static` `void` `main (String[] args)` `{` `    ``int` `A[] = { ``2``, ``3``, ``5` `}; ` `    ``int` `n = A.length; ` `    `  `    ``// Function call ` `    ``checkPairwiseCoPrime(A, n); ` `}` `}`   `// This code is contributed by offbeat`

 `# Python3 program for the above approach `   `# Function to calculate GCD ` `def` `GCD(a, b): ` `    `  `    ``if` `(a ``=``=` `0``): ` `        ``return` `b ` `        `  `    ``return` `GCD(b ``%` `a, a) `   `# Function to calculate LCM ` `def` `LCM(a, b): ` `    `  `    ``return` `(a ``*` `b) ``/``/` `GCD(a, b) `   `# Function to check if aelements ` `# in the array are pairwise coprime ` `def` `checkPairwiseCoPrime(A, n): ` `    `  `    ``# Initialze variables ` `    ``prod ``=` `1` `    ``lcm ``=` `1`   `    ``# Itertae over the array ` `    ``for` `i ``in` `range``(n): `   `        ``# Calculate product of ` `        ``# array elements ` `        ``prod ``*``=` `A[i] `   `        ``# Calculate LCM of ` `        ``# array elements ` `        ``lcm ``=` `LCM(A[i], lcm) `   `    ``# If the product of array elements ` `    ``# is equal to LCM of the array ` `    ``if` `(prod ``=``=` `lcm): ` `        ``print``(``"Yes"``) ` `    ``else``: ` `        ``print``(``"No"``) `   `# Driver Code ` `if` `__name__ ``=``=` `'__main__'``: ` `    `  `    ``A ``=` `[ ``2``, ``3``, ``5` `] ` `    ``n ``=` `len``(A) `   `    ``# Function call ` `    ``checkPairwiseCoPrime(A, n) `   `# This code is contributed by mohit kumar 29 `

 `// C# program for ` `// the above approach` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG{`   `// Function to calculate GCD ` `static` `long` `GCD(``long` `a, ` `                ``long` `b) ` `{ ` `  ``if` `(a == 0) ` `    ``return` `b; ` `  ``return` `GCD(b % a, a); ` `} `   `// Function to calculate LCM ` `static` `long` `LCM(``long` `a, ` `                ``long` `b) ` `{ ` `  ``return` `(a * b) / GCD(a, b); ` `} `   `// Function to check if all elements ` `// in the array are pairwise coprime ` `static` `void` `checkPairwiseCoPrime(``int` `[]A, ` `                                 ``int` `n) ` `{     ` `  ``// Initialze variables ` `  ``long` `prod = 1; ` `  ``long` `lcm = 1; `   `  ``// Itertae over the array ` `  ``for``(``int` `i = 0; i < n; i++)` `  ``{ ` `    ``// Calculate product of ` `    ``// array elements ` `    ``prod *= A[i]; `   `    ``// Calculate LCM of ` `    ``// array elements ` `    ``lcm = LCM(A[i], lcm); ` `  ``} `   `  ``// If the product of array elements ` `  ``// is equal to LCM of the array ` `  ``if` `(prod == lcm) ` `    ``Console.WriteLine(``"Yes"``); ` `  ``else` `    ``Console.WriteLine(``"No"``); ` `}`   `// Driver Code` `public` `static` `void` `Main(String[] args)` `{` `  ``int` `[]A = {2, 3, 5}; ` `  ``int` `n = A.Length; `   `  ``// Function call ` `  ``checkPairwiseCoPrime(A, n); ` `}` `}`   `// This code is contributed by Rajput-Ji`

Output:
```Yes

```

Time Complexity: O(N log (min(A[i])))
Auxiliary Space: O(1)

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Improved By : mohit kumar 29, offbeat, Rajput-Ji

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