Count Array Pairs Divisible by K

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from collections import defaultdict
from math import gcd

class Solution:
    def countPairs(self, nums: List[int], k: int) -> int:
        gcdCount = defaultdict(int)
        ans = 0

        for num in nums:
            currgcd = gcd(num, k)
            for gc_d, count in gcdCount.items():
                if (currgcd * gc_d) % k == 0:
                    ans += count
            gcdCount[currgcd] += 1

        return ans

Identifying Problem Isomorphism

“Count Array Pairs Divisible by K” can be approximately mapped to “Subarray Sums Divisible by K”.

Reasoning:

Both share a similar concept: to work with arrays and count elements or pairs whose sums are divisible by a given number ‘K’.

In “Count Array Pairs Divisible by K”, the task is to count pairs from an array whose sum is divisible by ‘K’.

“Subarray Sums Divisible by K” requires to count all the continuous subarrays whose sum is divisible by ‘K’.

Despite the difference in the level of granularity (pairs vs subarrays), both problems require similar techniques to solve - particularly using the modulo operation and hashing.

“Subarray Sums Divisible by K” is simpler as it works with subarrays that have a continuous range in the original array. “Count Array Pairs Divisible by K” is more complex because it does not restrict the pair formation to contiguous elements, thereby increasing the number of potential combinations to evaluate.

10 Prerequisite LeetCode Problems

This requires you to work with arrays and handle modulo operations. Here are 10 problems to prepare for this one:

  1. 1. Two Sum: A good starting point to learn about iterating over pairs in an array.

  2. 167. Two Sum II - Input array is sorted: This is similar to the first problem, but the array is sorted.

  3. 560. Subarray Sum Equals K: This problem will give you practice with working with sums of subarrays.

  4. 974. Subarray Sums Divisible by K: In this problem, you’ll need to count the number of subarrays that have a sum divisible by K.

  5. 523. Continuous Subarray Sum: Similar to the previous problem, you’ll need to find if there is a subarray with a sum that is a multiple of K.

  6. 49. Group Anagrams: This problem will give you practice with handling groups of elements with a particular property.

  7. 325. Maximum Size Subarray Sum Equals k: This is another problem about finding subarrays with a sum that equals to a given number.

  8. 229. Majority Element II: This problem will give you practice with counting occurrences of elements in an array.

  9. 1481. Least Number of Unique Integers after K Removals: This problem will give you practice with removing elements from an array.

  10. 15. 3Sum: This problem involves searching for triplets in the array that add up to a target number.

You will understand how to handle arrays, perform modulo operations, and work with pairs of elements.

Problem Classification

Problem Statement:Given a 0-indexed integer array nums of length n and an integer k, return the number of pairs (i, j) such that:

0 <= i < j <= n - 1 and nums[i] * nums[j] is divisible by k.

Example 1:

Input: nums = [1,2,3,4,5], k = 2 Output: 7 Explanation: 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 Explanation: 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

Analyze the provided problem statement. Categorize it based on its domain, ignoring ‘How’ it might be solved. Identify and list out the ‘What’ components. Based on these, further classify the problem. Explain your categorizations.

Clarification Questions

What are the clarification questions we can ask about this problem?

Identifying Problem Isomorphism

Can you help me with finding the isomorphism for this problem?

Which problem does this problem map to the corresponding isomorphic problem on Leetcode ?

Visual Model of the Problem

How to visualize the problem statement for this problem?

Problem Restatement

Could you start by paraphrasing the problem statement in your own words? Try to distill the problem into its essential elements and make sure to clarify the requirements and constraints. This exercise should aid in understanding the problem better and aligning our thought process before jumping into solving it.

Abstract Representation of the Problem

Could you help me formulate an abstract representation of this problem?

Given this problem, how can we describe it in an abstract way that emphasizes the structure and key elements, without the specific real-world details?

Terminology

Are there any specialized terms, jargon, or technical concepts that are crucial to understanding this problem or solution? Could you define them and explain their role within the context of this problem?

Problem Simplification and Explanation

Could you please break down this problem into simpler terms? What are the key concepts involved and how do they interact? Can you also provide a metaphor or analogy to help me understand the problem better?

Constraints

Given the problem statement and the constraints provided, identify specific characteristics or conditions that can be exploited to our advantage in finding an efficient solution. Look for patterns or specific numerical ranges that could be useful in manipulating or interpreting the data.

What are the key insights from analyzing the constraints?

Case Analysis

Could you please provide additional examples or test cases that cover a wider range of the input space, including edge and boundary conditions? In doing so, could you also analyze each example to highlight different aspects of the problem, key constraints and potential pitfalls, as well as the reasoning behind the expected output for each case? This should help in generating key insights about the problem and ensuring the solution is robust and handles all possible scenarios.

Identification of Applicable Theoretical Concepts

Can you identify any mathematical or algorithmic concepts or properties that can be applied to simplify the problem or make it more manageable? Think about the nature of the operations or manipulations required by the problem statement. Are there existing theories, metrics, or methodologies in mathematics, computer science, or related fields that can be applied to calculate, measure, or perform these operations more effectively or efficiently?

Problem Breakdown and Solution Methodology

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Inference of Problem-Solving Approach from the Problem Statement

How did you infer from the problem statement that this problem can be solved using ?

Stepwise Refinement

  1. Could you please provide a stepwise refinement of our approach to solving this problem?

  2. How can we take the high-level solution approach and distill it into more granular, actionable steps?

  3. Could you identify any parts of the problem that can be solved independently?

  4. Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Given the problem statement, can you explain in detail how you would approach solving it? Please break down the process into smaller steps, illustrating how each step contributes to the overall solution. If applicable, consider using metaphors, analogies, or visual representations to make your explanation more intuitive. After explaining the process, can you also discuss how specific operations or changes in the problem’s parameters would affect the solution? Lastly, demonstrate the workings of your approach using one or more example cases.

Thought Process

Explain the thought process by thinking step by step to solve this problem from the problem statement and code the final solution. Write code in Python3. What are the cues in the problem statement? What direction does it suggest in the approach to the problem? Generate insights about the problem statement.

From Brute Force to Optimal Solution

Could you please begin by illustrating a brute force solution for this problem? After detailing and discussing the inefficiencies of the brute force approach, could you then guide us through the process of optimizing this solution? Please explain each step towards optimization, discussing the reasoning behind each decision made, and how it improves upon the previous solution. Also, could you show how these optimizations impact the time and space complexity of our solution?

Coding Constructs

Consider the following piece of complex software code.

  1. What are the high-level problem-solving strategies or techniques being used by this code?

  2. If you had to explain the purpose of this code to a non-programmer, what would you say?

  3. Can you identify the logical elements or constructs used in this code, independent of any programming language?

  4. Could you describe the algorithmic approach used by this code in plain English?

  5. What are the key steps or operations this code is performing on the input data, and why?

  6. Can you identify the algorithmic patterns or strategies used by this code, irrespective of the specific programming language syntax?

Language Agnostic Coding Drills

Your mission is to deconstruct this code into the smallest possible learning units, each corresponding to a separate coding concept. Consider these concepts as unique coding drills that can be individually implemented and later assembled into the final solution.

  1. Dissect the code and identify each distinct concept it contains. Remember, this process should be language-agnostic and generally applicable to most modern programming languages.

  2. Once you’ve identified these coding concepts or drills, list them out in order of increasing difficulty. Provide a brief description of each concept and why it is classified at its particular difficulty level.

  3. Next, describe the problem-solving approach that would lead from the problem statement to the final solution. Think about how each of these coding drills contributes to the overall solution. Elucidate the step-by-step process involved in using these drills to solve the problem. Please refrain from writing any actual code; we’re focusing on understanding the process and strategy.

Targeted Drills in Python

Now that you’ve identified and ordered the coding concepts from a complex software code in the previous exercise, let’s focus on creating Python-based coding drills for each of those concepts.

  1. Begin by writing a separate piece of Python code that encapsulates each identified concept. These individual drills should illustrate how to implement each concept in Python. Please ensure that these are suitable even for those with a basic understanding of Python.

  2. In addition to the general concepts, identify and write coding drills for any problem-specific concepts that might be needed to create a solution. Describe why these drills are essential for our problem.

  3. Once all drills have been coded, describe how these pieces can be integrated together in the right order to solve the initial problem. Each drill should contribute to building up to the final solution.

Remember, the goal is to not only to write these drills but also to ensure that they can be cohesively assembled into one comprehensive solution.

Q&A

Similar Problems

Given the problem , identify and list down 10 similar problems on LeetCode. These should cover similar concepts or require similar problem-solving approaches as the provided problem. Please also give a brief reason as to why you think each problem is similar to the given problem.