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# Maximize size of Subset from Matrix values such that any pair is coprime

Given two integers N and M which denotes a matrix of size N*M that has the value of each cell as the sum of its row number and column number, the task is to find the maximum size of a subset that can be formed using the elements of this matrix such that any pair of the subset will be coprime to each other.

Examples:

Input: N = 3, M = 4
Output: 4
Explanation: There are a maximum of 4 possible numbers on the matrix, Which are: {2, 5, 7, 3}, This makes pairs with the combination itself and has the GCD of that pair equal to one. i, e. (2, 3), (2, 5), (5, 3) . . . The matrix is shown below

Input: N = 5, M = 8
Output: 6

Approach: The problem can be solved based on the following observation

As any pair of the subset will be coprime to each other, therefore the subset will be formed using all the prime within the range [2, N+M].

Follow the steps mentioned below to implement the idea:

• Create a function to check if a number is prime or not.
• Run a loop from i = 2 to N+M.
• Check whether the current number is prime or not.
• If yes then increment the counter variable.
• Last, return the count of primes as the answer.

Below is the Implementation of the above approach.

## C++

```// C++ code to implement the approach
#include <bits/stdc++.h>
using namespace std;

// Boolean function to check a number is prime or not
bool prime(int num)
{
for (int i = 2; i * i <= num; i++) {
if (num % i == 0) {
return false;
}
}
return true;
}

// Function to find the size of the subset
int cal(int n, int m)
{
// Variable to count pairs
int count = 0;

// Loop for traversing 2 to (N+M)
for (int i = 2; i <= (n + m); i++) {

// Checking if current number
// of matrix is prime or not
if (prime(i)) {

// Incrementing count variable
count++;
}
}

// Return maximum number of pairs
return count;
}

// Driver Code
int main()
{
int N = 3, M = 4;

// Function call
cout << cal(N, M);
return 0;
}

// This code is contributed by Rohit Pradhan```

## Java

```// Java code to implement the approach

import java.io.*;
import java.lang.*;
import java.util.*;

class GFG {

// Boolean function to check a number is prime or not
public static boolean prime(int num)
{
for (int i = 2; i * i <= num; i++) {
if (num % i == 0) {
return false;
}
}
return true;
}

// Function to find the size of the subset
public static int cal(int n, int m)
{
// Variable to count pairs
int count = 0;

// Loop for traversing 2 to (N+M)
for (int i = 2; i <= (n + m); i++) {

// Checking if current number
// of matrix is prime or not
if (prime(i)) {

// Incrementing count variable
count++;
}
}

// Return maximum number of pairs
return count;
}

// Driver Code
public static void main(String[] args)
{
int N = 3, M = 4;

// Function call
System.out.println(cal(N, M));
}
}```

## Python3

```# Python code for the above approach

# Function to check a number is prime or not
def prime(num):
for i in range(2, int(num**0.5)+1):
if num % i == 0:
return False
return True

# Function to find the size of the subset
def cal(n, m):
# Variable to count pairs
count = 0

# Loop for traversing 2 to (N+M)
for i in range(2, n+m+1):
# Checking if current number
# of matrix is prime or not
if prime(i):
# Incrementing count variable
count += 1

# Return maximum number of pairs
return count

# Driver Code
N = 3
M = 4

# Function call
print(cal(N, M))

# This code is contributed by lokesh.
```

## C#

```// C# code to implement the approach
using System;

class GFG {

// Boolean function to check a number is prime or not
public static bool prime(int num)
{
for (int i = 2; i * i <= num; i++) {
if (num % i == 0) {
return false;
}
}
return true;
}

// Function to find the size of the subset
public static int cal(int n, int m)
{
// Variable to count pairs
int count = 0;

// Loop for traversing 2 to (N+M)
for (int i = 2; i <= (n + m); i++) {

// Checking if current number
// of matrix is prime or not
if (prime(i)) {

// Incrementing count variable
count++;
}
}

// Return maximum number of pairs
return count;
}

// Driver Code
static public void Main()
{
int N = 3, M = 4;

// Function call
Console.Write(cal(N, M));
}
}

// This code is contributed by Pushpesh Raj.
```

## Javascript

```// Javascript code to implement the approach

// Boolean function to check a number is prime or not
function prime(num)
{
for (let i = 2; i * i <= num; i++) {
if (num % i == 0) {
return false;
}
}
return true;
}

// Function to find the size of the subset
function cal(n, m)
{
// Variable to count pairs
let count = 0;

// Loop for traversing 2 to (N+M)
for (let i = 2; i <= (n + m); i++) {

// Checking if current number
// of matrix is prime or not
if (prime(i)) {

// Incrementing count variable
count++;
}
}

// Return maximum number of pairs
return count;
}

// Driver Code
let N = 3, M = 4;

// Function call
document.write(cal(N, M));
```
Output

`4`

Time Complexity: O((N+M) * sqrt(N+M))
Auxiliary Space: O(1)

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