Given an array \(a\) of positive integers, find the number of pairs of integers for which the difference between the two numbers is equal to a given number \(k\). Since the number of pairs could be quite large, take it modulo \(10^9 + 7\) for your output.

Choose one of the following approaches:

A) Use a brute force approach to iterate through all possible pairs and count the pairs that satisfy the given condition.

B) Sort the array in ascending order and use two pointers to find pairs with the desired difference.

C) Create a hash set to store the elements of the array and iterate through the array to check if the complement of each element exists in the set.

D) Implement a dynamic programming solution to count the number of pairs with the desired difference.

The problem involves finding pairs of integers in an array with a specific difference. This can be solved using a hash set to store the array elements, then checking for each element's complement in the set. Alternatively, the array can be sorted and two pointers used to locate pairs with the desired difference.

Explanation:

The subject here involves finding the number of pairs of integers in an array whose difference is equal to a given number, k. The process involves mathematical computation heavily based on algorithms and data structures, which is more inclined towards Computers and Technology.

One approach to tackling the problem is to create a hash set to store the elements of the array. Then, you iterate through the array checking if the complement (difference from k) of each element exists in the set. For example, given an array S = {1, 2, 3, 4, 5... 19} and a given difference k, we can store the elements in a hash set, and for each element, check if its complement exists in the set.

Another approach would be to sort the array in ascending order and use two pointers to find pairs with the desired difference k. This would involve moving the pointers based on the difference of the elements they point to until you have moved through the whole array.

Note that for large arrays, the outcome may be quite large, thus the need to take it modulo 109 + 7 to limit the size of the output. Remember, the goal is to find the number of pairs with a difference that equals the given number, k.

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