Count Array Pairs Divisible by K

This problem involves finding pairs of indices in an array such that their product is divisible by a given integer k. The challenge lies in efficiently calculating these pairs while considering divisibility conditions. Understanding how to use number theory and array operations optimally is key to solving this problem.

Problem Statement

Given a 0-indexed integer array nums and an integer k, your task is to return the number of pairs (i, j) such that the product of nums[i] and nums[j] is divisible by k. Both i and j should satisfy 0 <= i < j < n, where n is the length of the array.

The problem asks you to consider how each element can form a pair with another, where the product of the selected elements is divisible by k. For large arrays, a brute force approach would not be efficient, so identifying a more optimal approach based on divisibility rules is crucial.

Examples

Example 1

Input: nums = [1,2,3,4,5], k = 2

Output: 7

The 7 pairs of indices whose corresponding products are divisible by 2 are (0, 1), (0, 3), (1, 2), (1, 3), (1, 4), (2, 3), and (3, 4). Their products are 2, 4, 6, 8, 10, 12, and 20 respectively. Other pairs such as (0, 2) and (2, 4) have products 3 and 15 respectively, which are not divisible by 2.

Example 2

Input: nums = [1,2,3,4], k = 5

Output: 0

There does not exist any pair of indices whose corresponding product is divisible by 5.

Constraints

• 1 <= nums.length <= 105
• 1 <= nums[i], k <= 105

Solution Approach

Divisibility Condition

For each element in the array, determine the smallest number it should be multiplied by to ensure the product is divisible by k. This involves understanding divisibility rules and can be optimized with number theory techniques, avoiding direct brute force checks for each pair.

Frequency Count Optimization

By using a frequency count of the remainders when elements are divided by k, we can efficiently pair elements with compatible remainders to form divisible products. This approach reduces the complexity from O(n^2) to O(n).

Modulo Arithmetic

Modulo arithmetic plays a key role in this problem. For each pair, the product must satisfy the condition that the sum of the remainders modulo k equals zero. This insight helps narrow down which pairs to check, streamlining the solution.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the final approach. Using a frequency-based method to pair numbers efficiently leads to O(n) time complexity, where n is the length of the array. Space complexity is also O(n) due to the storage of frequency counts or modulo operations.

What Interviewers Usually Probe

• Ability to optimize brute force approaches.
• Understanding of number theory and modular arithmetic.
• Familiarity with efficient frequency counting techniques.

Common Pitfalls or Variants

Common pitfalls

• Using a brute force approach without optimizing divisibility checks.
• Overlooking how modular arithmetic can simplify the solution.
• Not considering edge cases where no valid pairs exist, like when the product is never divisible by k.

Follow-up variants

• What if k is a prime number?
• How would the solution change if we wanted to count pairs whose product is divisible by a prime factor of k?
• Can this solution be adapted for non-zero index-based array pairs?

FAQ

What is the optimal approach for solving Count Array Pairs Divisible by K?

The optimal approach involves using number theory and frequency counting of remainders modulo k to efficiently find pairs that satisfy the divisibility condition.

How can modular arithmetic simplify the solution for this problem?

Modular arithmetic helps identify pairs where the sum of their remainders modulo k equals zero, streamlining the solution and avoiding brute force checking.

What is the time complexity of the optimal solution for Count Array Pairs Divisible by K?

The time complexity of the optimal solution is O(n), where n is the length of the array, due to the use of frequency counts and modular arithmetic.

Are there any edge cases I should consider for this problem?

Edge cases include situations where no valid pairs exist, such as when the product of any pair in the array is never divisible by k.

What is the role of frequency counting in solving this problem?

Frequency counting helps track how many elements in the array have specific remainders when divided by k, which simplifies the process of finding divisible pairs.