Valid Pairs Problem Statement

Given an array of integers ARR with a size of 'N' and two integers 'K' and 'M', determine if it is possible to divide the array into pairs such that the sum of every pair yields a remainder 'M' when divided by 'K'. If possible, return true; otherwise, return false.

Example:

ARR = [2, 1, 5, 7], K = 9, M = 3

Output:

true

Explanation:

The array can be divided into pairs (2, 1) and (5, 7), whose sums are 3 and 12, respectively. Both sums give a remainder of 3 when divided by 9.

Constraints:

• 1 <= T <= 10
• 1 <= N <= 10 ^ 5
• 1 <= ARR[i] <= 10 ^ 9
• 1 <= K <= 10 ^ 9
• 0 <= M < K

Input:

The first line of input contains a single integer 'T', representing the number of test cases or queries to be run. The 'T' test cases follow. The first line of each test case contains an integer 'N' representing the size of the given array. The second line contains 'N' single space-separated integers representing the elements of the array 'ARR'. The third line contains two single space-separated integers 'K' and 'M'.

Output:

For each test case, print a single line containing either "True" or "False".

Note:

Every element of the array should contribute to only one pair, and you do not need to handle output printing as it is already managed by the system. Implement only the given function logic.