Help me out with this question

general
1

you are given a array of positive integers a and a number x print number of pairs such that i<j and a[i]==a[j] and i*j should be divisible by x
(solved in o(n2)) need an optimal solution
array indexing 1 based
example=[2,2,2] x=2
arr[i]=2 for all valid i and the valid indices are [1,2] and [2,3]
so output should be 2

2

which language are you using?

3

Absolutely, there’s a more efficient approach to solve this problem with a time complexity of O(N) compared to the O(N^2) solution using nested loops. Here’s how we can achieve this:

  1. Frequency Count:

    • Initialize a dictionary (or hash map) to store the frequency of each element in the array a.
    • Iterate through the array and update the frequency count for each element.

  2. Divisibility Analysis:

    • Iterate through the frequency dictionary.
    • For each element (num) and its frequency (num_freq):
      • Check if num * num (square of the element) is divisible by x.
        • If yes, this contributes num_freq * (num_freq - 1) // 2 pairs. This follows the combination formula to calculate the number of ways to choose two elements from num_freq elements (considering order doesn’t matter).
      • Check if x % num == 0 (i.e., x is divisible by num) and num != x // num (to avoid double counting when x is a perfect square of num).
        • If yes, this contributes another freq[x // num] * num_freq pairs. This is because for every element num, there exists another element x // num (assuming x is divisible by num) such that their product is divisible by x. We multiply the frequency of x // num with num_freq to count the pairs.

  3. Return the Count:

    • After iterating through the frequency dictionary, return the total count of valid pairs.

This approach leverages the frequency information to efficiently identify potential pairs without nested loops.