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Count of pairs in an array such that the highest power of 2 that divides their product is 1

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  • Last Updated : 28 May, 2022
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Given an array arr[] of N positive integers. The task is to find the count of pairs (arr[i], arr[j]) such that the maximum power of 2 that divides arr[i] * arr[j] is 1.
Examples: 
 

Input: arr[] = {3, 5, 2, 8} 
Output:
(3, 2), (5, 2) and (3, 5) are the only valid pairs. 
Since the power of 2 that divides 3 * 2 = 6 is 1, 
5 * 2 = 10 is 1 and 3 * 5 = 15 is 0.
Input: arr[] = {4, 2, 7, 11, 14, 15, 18} 
Output: 12 
 

 

Approach: As the maximum power of 2 that divides arr[i] * arr[j] is at max 1, it means that if P is the product then it must either be odd or 2 is the only even factor of P
It implies that both arr[i] and arr[j] must be odd or exactly one of them is even and 2 is the only even factor of this number. 
If odd is the count of odd numbers and even is the count of even numbers such that 2 is the only even factor of that number then the answer will be odd * even + odd * (odd – 1) / 2.
Below is the implementation of the above approach: 
 

C++




// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to return the count of valid pairs
int cntPairs(int a[], int n)
{
 
    // To store the count of odd numbers and
    // the count of even numbers such that 2
    // is the only even factor of that number
    int odd = 0, even = 0;
 
    for (int i = 0; i < n; i++) {
 
        // If current number is odd
        if (a[i] % 2 == 1)
            odd++;
 
        // If current number is even and 2
        // is the only even factor of it
        else if ((a[i] / 2) % 2 == 1)
            even++;
    }
 
    // Calculate total number of valid pairs
    int ans = odd * even + (odd * (odd - 1)) / 2;
 
    return ans;
}
 
// Driver code
int main()
{
 
    int a[] = { 4, 2, 7, 11, 14, 15, 18 };
    int n = sizeof(a) / sizeof(a[0]);
 
    cout << cntPairs(a, n);
 
    return 0;
}

Java




// Java implementation of the approach
class GFG
{
 
// Function to return the count of valid pairs
static int cntPairs(int a[], int n)
{
 
    // To store the count of odd numbers and
    // the count of even numbers such that 2
    // is the only even factor of that number
    int odd = 0, even = 0;
 
    for (int i = 0; i < n; i++)
    {
 
        // If current number is odd
        if (a[i] % 2 == 1)
            odd++;
 
        // If current number is even and 2
        // is the only even factor of it
        else if ((a[i] / 2) % 2 == 1)
            even++;
    }
 
    // Calculate total number of valid pairs
    int ans = odd * even + (odd * (odd - 1)) / 2;
 
    return ans;
}
 
// Driver code
public static void main(String []args)
{
    int a[] = { 4, 2, 7, 11, 14, 15, 18 };
    int n = a.length;
 
    System.out.println(cntPairs(a, n));
}
}
 
// This code is contributed by 29AjayKumar

Python3




# Python3 implementation of the approach
 
# Function to return the count of valid pairs
def cntPairs(a, n) :
 
    # To store the count of odd numbers and
    # the count of even numbers such that 2
    # is the only even factor of that number
    odd = 0; even = 0;
 
    for i in range(n) :
 
        # If current number is odd
        if (a[i] % 2 == 1) :
            odd += 1;
 
        # If current number is even and 2
        # is the only even factor of it
        elif ((a[i] / 2) % 2 == 1) :
            even += 1;
     
    # Calculate total number of valid pairs
    ans = odd * even + (odd * (odd - 1)) // 2;
 
    return ans;
 
# Driver code
if __name__ == "__main__" :
 
    a = [ 4, 2, 7, 11, 14, 15, 18 ];
    n = len(a);
 
    print(cntPairs(a, n));
     
# This code is contributed by AnkitRai01

C#




// C# implementation of the approach
using System;
     
class GFG
{
 
// Function to return the count of valid pairs
static int cntPairs(int []a, int n)
{
 
    // To store the count of odd numbers and
    // the count of even numbers such that 2
    // is the only even factor of that number
    int odd = 0, even = 0;
 
    for (int i = 0; i < n; i++)
    {
 
        // If current number is odd
        if (a[i] % 2 == 1)
            odd++;
 
        // If current number is even and 2
        // is the only even factor of it
        else if ((a[i] / 2) % 2 == 1)
            even++;
    }
 
    // Calculate total number of valid pairs
    int ans = odd * even + (odd * (odd - 1)) / 2;
 
    return ans;
}
 
// Driver code
public static void Main(String []args)
{
    int []a = { 4, 2, 7, 11, 14, 15, 18 };
    int n = a.Length;
 
    Console.WriteLine(cntPairs(a, n));
}
}
 
// This code is contributed by Ajay KUmar

Javascript




<script>
 
// Javascript implementation of the approach
 
// Function to return the count of valid pairs
function cntPairs(a, n)
{
 
    // To store the count of odd numbers and
    // the count of even numbers such that 2
    // is the only even factor of that number
    var odd = 0, even = 0;
 
    for (var i = 0; i < n; i++) {
 
        // If current number is odd
        if (a[i] % 2 == 1)
            odd++;
 
        // If current number is even and 2
        // is the only even factor of it
        else if ((a[i] / 2) % 2 == 1)
            even++;
    }
 
    // Calculate total number of valid pairs
    var ans = odd * even + (odd * (odd - 1)) / 2;
 
    return ans;
}
 
// Driver code
var a = [4, 2, 7, 11, 14, 15, 18];
var n = a.length;
document.write( cntPairs(a, n));
 
// This code is contributed by rrrtnx.
</script>

Output: 

12

 

Time Complexity: O(n) where n is number of elements in given array. As, we are using a loop to traverse N times so it will cost us O(N) time 
Auxiliary Space: O(1), as we are not using any extra space.
 


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