Check If Array Pairs Are Divisible by k

Problem Description

Given an array of integers arr of even length n and an integer k, we want to divide the array into exactly n / 2 pairs such that the sum of each pair is divisible by k. Return true if you can find a way to do that or false otherwise.

Key Insights

• Each element in the array can be associated with a remainder when divided by k.
• For two numbers to form a valid pair, the sum of their remainders must also be divisible by k.
• We can count occurrences of each remainder, and then match them appropriately to form pairs.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(k)

Solution

To solve the problem, we can use a hash table (or array) to count the occurrences of each remainder when the elements of the array are divided by k. The key steps in the algorithm are:

1. Iterate through the array and compute the remainder of each element when divided by k.
2. Maintain a count of how many times each remainder appears.
3. Check if we can form valid pairs based on the counts of remainders:
   • Remainders that are 0 must be paired among themselves.
   • Remainders that are exactly half of k must also be paired among themselves.
   • For other remainders, ensure that the count of a remainder matches the count of its complement (k - remainder).

Code Solutions