Number of Subarrays With GCD Equal to K

To solve this problem, we need to understand the properties of the greatest common divisor (GCD). For each subarray of the array nums, we have to check if its GCD is k. Let’s break down the problem step by step.

Observations

For any subarray, if its GCD is k, then all its elements must be divisible by k. Otherwise, the GCD of the subarray would be less than k.
Checking for divisibility can be an efficient way to skip unnecessary subarrays. However, even if all numbers in a subarray are divisible by k, it doesn’t guarantee that the GCD of the entire subarray is k.

Algorithm

Initialize a counter to 0 to count subarrays.
For each starting position i of the array nums: a. For each ending position j greater than or equal to i: i. Check if nums[j] is divisible by k. If not, break the loop as this subarray and all the longer subarrays starting from i cannot have a GCD of k. ii. Compute the GCD of the subarray from i to j. If the GCD is k, increment the counter.
Return the counter.

Implementation

To efficiently compute the GCD of subarrays, we’ll use the Euclidean algorithm:

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def gcd(a: int, b: int) -> int:
    while b:
        a, b = b, a % b
    return a

Using the above GCD function, the solution becomes:

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class Solution:
    def subarrayGCD(self, nums: List[int], k: int) -> int:
        n = len(nums)
        count = 0

        for i in range(n):
            curr_gcd = 0
            for j in range(i, n):
                if nums[j] % k != 0:
                    break
                curr_gcd = gcd(curr_gcd, nums[j])
                if curr_gcd == k:
                    count += 1

        return count

Key Takeaways

Utilizing properties of the GCD helps in pruning the search space.
The Euclidean algorithm is an efficient way to compute the GCD of two numbers. In this problem, we iteratively compute the GCD of subarrays using this algorithm.

10 Prerequisite LeetCode Problems

For this, the following are a good preparation:

“560. Subarray Sum Equals K”: This problem involves identifying subarrays with a specific sum, which requires similar skills to identifying subarrays with a specific GCD.
“209. Minimum Size Subarray Sum”: The task is to find a continuous subarray which adds up to a specific value, similar to finding a subarray with a particular GCD in the target problem.
“713. Subarray Product Less Than K”: This problem helps understand how to handle subarrays and their products, which can be translated to dealing with GCDs.
“974. Subarray Sums Divisible by K”: Learning to handle the property of divisibility in this problem will help with understanding how to handle GCDs.
“1073. Adding Two Negabinary Numbers”: This problem is about binary number operations and can help understanding operations with integers, which will be beneficial when working with GCDs.
“930. Binary Subarrays With Sum”: This problem helps understand the concept of subarrays with a particular sum, which can be adapted to handle subarrays with a particular GCD.
“1480. Running Sum of 1d Array”: This problem can provide a foundation for understanding how to calculate continuous sums, which is relevant for the continuous GCD in the problem.
“525. Contiguous Array”: This problem involves finding a maximum length subarray with equal number of 0 and 1. The concept can be somewhat translated to finding subarrays with specific GCD.
“907. Sum of Subarray Minimums”: This problem asks for the sum of the minimums of all subarrays, which is a step toward understanding GCDs of all subarrays.
“303. Range Sum Query - Immutable”: Understanding how to efficiently calculate the sum of elements in a specific range of an array will help to grasp the similar concept of calculating GCD for a specific range of subarrays.

These cover identifying subarrays, calculating sums or products of subarray elements, handling specific conditions within subarrays, and working with Greatest Common Divisor (GCD), which are relevant to solving the “2447. Number of Subarrays With GCD Equal to K” problem.

Problem Classification

Problem Statement:Given an integer array nums and an integer k, return the number of subarrays of nums where the greatest common divisor of the subarray’s elements is k. A subarray is a contiguous non-empty sequence of elements within an array. The greatest common divisor of an array is the largest integer that evenly divides all the array elements.

Example 1:

Input: nums = [9,3,1,2,6,3], k = 3 Output: 4 Explanation: The subarrays of nums where 3 is the greatest common divisor of all the subarray’s elements are:

[9,3,1,2,6,3]
[9,3,1,2,6,3]
[9,3,1,2,6,3]
[9,3,1,2,6,3]

Example 2:

Input: nums = [4], k = 7 Output: 0 Explanation: There are no subarrays of nums where 7 is the greatest common divisor of all the subarray’s elements.

Constraints:

1 <= nums.length <= 1000 1 <= nums[i], k <= 109

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?

Problem Analysis and Key Insights

What are the key insights from analyzing the problem statement?

Problem Boundary

What is the scope of this problem?

How to establish the boundary of this problem?

Distilling the Problem to Its Core Elements

Can you identify the fundamental concept or principle this problem is based upon? Please explain. What is the simplest way you would describe this problem to someone unfamiliar with the subject? What is the core problem we are trying to solve? Can we simplify the problem statement? Can you break down the problem into its key components? What is the minimal set of operations we need to perform to solve this problem?

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.

Provide names by categorizing these cases

What are the edge cases?

How to visualize these cases?

What are the key insights from analyzing the different cases?

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?

Simple Explanation

Can you explain this problem in simple terms or like you would explain to a non-technical person? Imagine you’re explaining this problem to someone without a background in programming. How would you describe it? If you had to explain this problem to a child or someone who doesn’t know anything about coding, how would you do it? In layman’s terms, how would you explain the concept of this problem? Could you provide a metaphor or everyday example to explain the idea of this problem?

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

Can you identify the key terms or concepts in this problem and explain how they inform your approach to solving it? Please list each keyword and how it guides you towards using a specific strategy or method. How can I recognize these properties by drawing tables or diagrams?

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

Simple Explanation of the Proof

I’m having trouble understanding the proof of this algorithm. Could you explain it in a way that’s easy to understand?

Could you please provide a stepwise refinement of our approach to solving this problem?
How can we take the high-level solution approach and distill it into more granular, actionable steps?
Could you identify any parts of the problem that can be solved independently?
Are there any repeatable patterns within our solution?

Solution Approach and Analysis

Identify Invariant

What is the invariant in this problem?

Identify Loop Invariant

What is the loop invariant in this problem?

Is invariant and loop invariant the same for this problem?

Thought Process

Can you explain the basic thought process and steps involved in solving this type of problem?

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.

Establishing Preconditions and Postconditions

Parameters:
- What are the inputs to the method?
- What types are these parameters?
- What do these parameters represent in the context of the problem?
Preconditions:
- Before this method is called, what must be true about the state of the program or the values of the parameters?
- Are there any constraints on the input parameters?
- Is there a specific state that the program or some part of it must be in?
Method Functionality:
- What is this method expected to do?
- How does it interact with the inputs and the current state of the program?
Postconditions:
- After the method has been called and has returned, what is now true about the state of the program or the values of the parameters?
- What does the return value represent or indicate?
- What side effects, if any, does the method have?
Error Handling:
- How does the method respond if the preconditions are not met?
- Does it throw an exception, return a special value, or do something else?

Problem Decomposition

Problem Understanding:
- Can you explain the problem in your own words? What are the key components and requirements?
Initial Breakdown:
- Start by identifying the major parts or stages of the problem. How can you break the problem into several broad subproblems?
Subproblem Refinement:
- For each subproblem identified, ask yourself if it can be further broken down. What are the smaller tasks that need to be done to solve each subproblem?
Task Identification:
- Within these smaller tasks, are there any that are repeated or very similar? Could these be generalized into a single, reusable task?
Task Abstraction:
- For each task you’ve identified, is it abstracted enough to be clear and reusable, but still makes sense in the context of the problem?
Method Naming:
- Can you give each task a simple, descriptive name that makes its purpose clear?
Subproblem Interactions:
- How do these subproblems or tasks interact with each other? In what order do they need to be performed? Are there any dependencies?

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?

Code Explanation and Design Decisions

Identify the initial parameters and explain their significance in the context of the problem statement or the solution domain.
Discuss the primary loop or iteration over the input data. What does each iteration represent in terms of the problem you’re trying to solve? How does the iteration advance or contribute to the solution?
If there are conditions or branches within the loop, what do these conditions signify? Explain the logical reasoning behind the branching in the context of the problem’s constraints or requirements.
If there are updates or modifications to parameters within the loop, clarify why these changes are necessary. How do these modifications reflect changes in the state of the solution or the constraints of the problem?
Describe any invariant that’s maintained throughout the code, and explain how it helps meet the problem’s constraints or objectives.
Discuss the significance of the final output in relation to the problem statement or solution domain. What does it represent and how does it satisfy the problem’s requirements?

Remember, the focus here is not to explain what the code does on a syntactic level, but to communicate the intent and rationale behind the code in the context of the problem being solved.

Coding Constructs

Consider the code for the solution of this problem.

What are the high-level problem-solving strategies or techniques being used by this code?
If you had to explain the purpose of this code to a non-programmer, what would you say?
Can you identify the logical elements or constructs used in this code, independent of any programming language?
Could you describe the algorithmic approach used by this code in plain English?
What are the key steps or operations this code is performing on the input data, and why?
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.

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.
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.
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.

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.
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.
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.