What is the time complexity of Split the Array to Make Coprime Products?

The optimal solution has a time complexity of O(N*log(MAX_VAL)).

What is the space complexity of Split the Array to Make Coprime Products?

The space complexity is O(MAX_VAL).

What data structures are used to solve Split the Array to Make Coprime Products?

This problem uses Array, Hash Table, Math, Number Theory. Understanding these concepts is essential for solving this Hard problem efficiently.

What are the key insights for solving Split the Array to Make Coprime Products?

Insight 1: We don't need to calculate the actual products. Instead, we can focus on the prime factors of each number. Two products are coprime if they share no common prime factors. Insight 2: Pre-computing the smallest prime factor (SPF) for all numbers up to the maximum value in `nums` allows us to efficiently find the prime factors of each number.

Which companies ask LeetCode 2584. Split the Array to Make Coprime Products in interviews?

This problem is frequently asked at Zomato.

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Split the Array to Make Coprime Products - LeetCode 2584 Solution

Q: How to solve Split the Array to Make Coprime Products?

The solution leverages prime factorization and a greedy approach. First, it pre-computes the smallest prime factor (SPF) for each number up to MAX_VAL. Then, for each element in `nums`, it finds the set of its prime factors. It stores the last occurrence of each prime factor in a hash map. Finally, it iterates through the array, maintaining a `max_reach`. For each index `i`, `max_reach` tracks the furthest index any prime factor seen up to `i` has occurred. If `max_reach` is equal to `i`, it means that all prime factors seen up to index `i` do not appear in the subarray `nums[i+1...]`, indicating that the product of `nums[0...i]` is coprime with the product of `nums[i+1...]`. Therefore `i` is a valid split point.

Split the Array to Make Coprime Products - Complete Solution Guide

Split the Array to Make Coprime Products is LeetCode problem 2584, a Hard level challenge. This complete guide provides step-by-step explanations, multiple solution approaches, and optimized code in python3, java, cpp, c.

Problem Statement

You are given a 0-indexed integer array nums of length n . A split at an index i where 0 <= i <= n - 2 is called valid if the product of the first i + 1 elements and the product of the remaining elements are coprime. For example, if nums = [2, 3, 3] , then a split at the index i = 0 is valid because 2 and 9 are coprime, while a split at the index i = 1 is not valid because 6 and 3 are not coprime. A split at the index i = 2 is not valid because i == n - 1 . Return the smallest index i at which t

Detailed Explanation

The problem asks us to find the smallest index in an array `nums` where we can split the array into two parts such that the product of the elements in the first part and the product of the elements in the second part are coprime. Two numbers are coprime if their greatest common divisor (GCD) is 1. If no such split exists, return -1.

Solution Approach

The solution leverages prime factorization and a greedy approach. First, it pre-computes the smallest prime factor (SPF) for each number up to MAX_VAL. Then, for each element in `nums`, it finds the set of its prime factors. It stores the last occurrence of each prime factor in a hash map. Finally, it iterates through the array, maintaining a `max_reach`. For each index `i`, `max_reach` tracks the furthest index any prime factor seen up to `i` has occurred. If `max_reach` is equal to `i`, it means that all prime factors seen up to index `i` do not appear in the subarray `nums[i+1...]`, indicating that the product of `nums[0...i]` is coprime with the product of `nums[i+1...]`. Therefore `i` is a valid split point.

Step-by-Step Algorithm

Step 1: Initialize an array `spf` of size MAX_VAL such that `spf[i]` stores the smallest prime factor of `i`.
Step 2: Populate the `spf` array using the Sieve of Eratosthenes. This allows for efficient prime factorization later.
Step 3: Create a function `get_prime_factors` that takes an integer as input and returns a set of its prime factors using the pre-computed `spf` array.
Step 4: Create a list `factors_list` where `factors_list[i]` is the set of prime factors of `nums[i]`.
Step 5: Create a hash map `last_occurrence` to store the last index at which each prime factor appears in `nums`.
Step 6: Iterate through `nums` up to `n - 2`. For each index `i`, update `max_reach` to be the maximum of its current value and the last occurrence of any prime factor of `nums[i]`. `max_reach` essentially represents the latest index any of the prime factors encountered so far have been seen.
Step 7: If `max_reach` is equal to `i`, it indicates that the product of elements up to index `i` is coprime with the product of elements from index `i+1` to the end of the array. In this case, return `i`.
Step 8: If the loop completes without finding a valid split point, return -1.

Key Insights

Insight 1: We don't need to calculate the actual products. Instead, we can focus on the prime factors of each number. Two products are coprime if they share no common prime factors.
Insight 2: Pre-computing the smallest prime factor (SPF) for all numbers up to the maximum value in `nums` allows us to efficiently find the prime factors of each number.
Insight 3: Keeping track of the last occurrence of each prime factor helps determine the furthest point we need to extend our left subarray to ensure coprimality with the remaining subarray.

Complexity Analysis

Time Complexity: O(N*log(MAX_VAL))

Space Complexity: O(MAX_VAL)

Topics

This problem involves: Array, Hash Table, Math, Number Theory.

Companies

Asked at: Zomato.

Split the Array to Make Coprime Products | LeetCode 2584

Hard

Array Hash Table Math Number Theory

Time: O(N*log(MAX_VAL))Space: O(MAX_VAL)

Solution Code

Available in:

import java.util.*;

class Solution {
    public int findValidSplit(int[] nums) {
        int MAX_VAL = 1000001;
        int[] spf = new int[MAX_VAL];
        for (int i = 0; i < MAX_VAL; i++) {
            spf[i] = i;
        }
        int i = 2;
        while (i * i < MAX_VAL) {
            if (spf[i] == i) {
                for (int j = i * i; j < MAX_VAL; j += i) {
                    if (spf[j] == j) {
                        spf[j] = i;
                    }
                }
            }
            i++;
        }

        
        
        
        
        
        
        
        
        
        
        

        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        

        
        

        
        

        
        
        
        
        

        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        Set<Set<Integer>> primeFactorSets = new HashSet<>();

View Full Solution on GitHubGitHub Solution View on LeetCodeLeetCode

Problem Info

Description

Problem 2584: Split the Array to Make Coprime Products - Detailed explanation and solution approaches.

Topics

Array Hash Table Math Number Theory

Companies

Zomato

Stats

Difficulty:Hard

Languages:4

Companies:1

Problem #:2584

Problem Explanation

💡

Think Before You Code

🤔 Ask Yourself

→What is the input? What output do I need?
→Can I solve a smaller version of this problem first?
→What patterns have I seen in similar problems?

✏️ Try This First

Before looking at the solution, try to solve it on paper with a simple example!

🚀 Try on LeetCode

Solution Approach

📋

Step-by-Step Solution

8 steps

1
Step 1: Initialize an array `spf` of size MAX_VAL such that `spf[i]` stores the smallest prime factor of `i`.
2
Step 2: Populate the `spf` array using the Sieve of Eratosthenes. This allows for efficient prime factorization later.
3
Step 3: Create a function `get_prime_factors` that takes an integer as input and returns a set of its prime factors using the pre-computed `spf` array.
4
Step 4: Create a list `factors_list` where `factors_list[i]` is the set of prime factors of `nums[i]`.
5
Step 5: Create a hash map `last_occurrence` to store the last index at which each prime factor appears in `nums`.
6
Step 6: Iterate through `nums` up to `n - 2`. For each index `i`, update `max_reach` to be the maximum of its current value and the last occurrence of any prime factor of `nums[i]`. `max_reach` essentially represents the latest index any of the prime factors encountered so far have been seen.
7
Step 7: If `max_reach` is equal to `i`, it indicates that the product of elements up to index `i` is coprime with the product of elements from index `i+1` to the end of the array. In this case, return `i`.
8
Step 8: If the loop completes without finding a valid split point, return -1.

💡

Pro Tip for Students

Try tracing through each step with a small example on paper before coding. Understanding "why" each step happens is more important than memorizing the code!

⚡

Complexity Analysis

⏱️

Time Complexity

O(N*log(MAX_VAL))

💾

Space Complexity

O(MAX_VAL)

📝

Why?

The `spf` array stores the smallest prime factor for each number up to MAX_VAL, resulting in O(MAX_VAL) space. The `last_occurrence` hash map stores the last index of each prime factor, so in the worst-case scenario where each number in the input array is a distinct prime number, the `last_occurrence` map could also take O(MAX_VAL) space. Other data structures such as 'factors_list' would take O(N*log(MAX_VAL)) space.

📖New to Big O? Click to learn more

Big O Notation - Quick Guide

O(1)Best - Instant

O(log n)Excellent - Binary Search

O(n)Good - One loop

O(n log n)OK - Sorting

O(n²)Slow - Nested loops

O(2ⁿ)Very Slow - Recursion

Alternative Approaches

Brute Force with GCD

Calculate the product of the first i+1 elements and the remaining elements for each possible split point. Then, compute the GCD of the two products. If the GCD is 1, we have a valid split. Naive GCD calculation can be very slow.

Time: O(N^2 * log(Product)), where N is the number of elements, Product is the overall product, because we have to calculate the product for each prefix/suffix which takes O(N) for each element and GCD is O(log(a*b)) where a,b are products of prefix/suffix.Space: O(1), as we only need to store the products and the GCD.

Trade-offs: This approach is simpler to understand but much less efficient, especially for larger input arrays and larger numbers. Not efficient enough for the given constraints.

Optimized GCD Approach

Instead of calculating products directly, iterate through the array, and for each possible split, determine if the left and right subarrays have any common prime factors. This can be done by keeping a count of prime factors for the left subarray and checking if any of those factors exist in the right subarray. A GCD function based on prime factorization might slightly improve efficiency.

Time: O(N * MAX_VAL), where MAX_VAL is the maximum value in the input array.Space: O(MAX_VAL) to store prime counts.

Trade-offs: Still less efficient than the provided solution because of the MAX_VAL in time complexity.

✨

Key Insights

1Insight 1: We don't need to calculate the actual products. Instead, we can focus on the prime factors of each number. Two products are coprime if they share no common prime factors.
2Insight 2: Pre-computing the smallest prime factor (SPF) for all numbers up to the maximum value in `nums` allows us to efficiently find the prime factors of each number.
3Insight 3: Keeping track of the last occurrence of each prime factor helps determine the furthest point we need to extend our left subarray to ensure coprimality with the remaining subarray.

📚

Key Terms Explained

For Students

Array

A collection of items stored at contiguous memory locations. Like a row of boxes.

Hash Table

A data structure that maps keys to values for quick lookup. Like a dictionary!

Edge Cases

Edge case 1: The array contains only one element. This is not a valid input based on the problem's constraint that `n >= 1` and the need for a split, so the code doesn't explicitly need to handle it, and the main logic will not be executed.
Edge case 2: The array cannot be split into two coprime products. The algorithm will iterate through the entire array and return -1 if no suitable split is found.
Edge case 3: All elements in `nums` are 1. This will cause the factors_list to contain only empty sets, and the code will return -1 as there will be no index `i` for which the two products are coprime since gcd(1,1) == 1

⚠️

Common Mistakes to Avoid

✗Common mistake 1: Calculating the actual products of the subarrays can lead to integer overflow, especially with the given constraints (1 <= nums[i] <= 10^6). Using prime factorization avoids this.

✗Common mistake 2: Not pre-computing the SPF leads to repeated calculations of prime factors, significantly increasing the time complexity.

✗Common mistake 3: Not using a hash map (or equivalent) to store the last occurrences of prime factors makes it difficult to efficiently determine the max_reach, resulting in a higher time complexity.

✗Common mistake 4: Incorrect handling of prime factors or using inefficient GCD algorithms can lead to the solution exceeding the time limit.

✅

Learning Tip

Every mistake is a learning opportunity! When you make an error, understand why it happened before moving on.

🌍

Where is This Used in Real Life?

💼Real-world scenario 1: Data partitioning in distributed systems where data should be divided into subsets that are relatively independent to minimize inter-partition communication.

🏦Real-world scenario 2: Cryptography, where coprime numbers play a fundamental role in generating keys and ensuring secure communication (e.g., in RSA algorithm).

💡Understanding real-world applications helps you see the "why" behind algorithms. It's not just about passing interviews—these concepts power the apps and websites you use every day!

Author's Notes

💡 Key Insight

When dealing with arrays, always consider if sorting would simplify the problem. Many array problems have O(n) solutions using hash maps or two-pointer technique.

🎯 Interview Tip

It's okay to not finish hard problems in interviews. Show your problem-solving process.

⚠️ Watch Out

Off-by-one errors are the most common mistake. Double-check your loop bounds and array indices.

Written by Saswata Mondal

Software Engineer • Algorithms Enthusiast • Interview Coach

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About the Author

👨‍💻

Saswata MondalAuthor400+ LC Problems

B.Tech CSE, IEM Kolkata | LeetCode Rating: 1672 (Top 15%)

Certified: IBM Java Developer, NPTEL Programming in Java

Last updated: August 4, 2024LeetCode Profile GitHub

This solution has been crafted based on my experience solving 400+ LeetCode problems. Each explanation is designed to help you understand the underlying concepts, not just memorize the code.

Community Tips & Insights

In a real interview, always clarify constraints first. Ask about input size and value ranges.

@TechLeadJul 14

Don't forget to handle the edge case when the input is empty. Many candidates miss this in interviews.

@DevExpert92Jul 7

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Frequently Asked Questions

What is the most efficient way to solve array problems?

Most array problems can be solved efficiently using techniques like two-pointer approach, sliding window, or hash maps. The key is identifying whether the array is sorted (enabling binary search) or if tracking seen elements (using hash maps) can help achieve O(n) time complexity.

What are the trade-offs of using a hash table?

Hash tables provide O(1) average-case lookup, insertion, and deletion, making them ideal for frequency counting and duplicate detection. However, they use O(n) extra space and may degrade to O(n) time in worst-case scenarios with hash collisions. Always consider if the space trade-off is acceptable.

How do I approach hard algorithm problems?

Hard problems often require: 1) Breaking down into subproblems, 2) Combining multiple advanced techniques, 3) Careful optimization of brute force. Start with a working solution (even O(n²) or worse), then optimize. Draw examples, identify patterns, and think about what makes this problem unique.

How should I practice algorithm problems effectively?

Focus on understanding patterns rather than memorizing solutions. After solving a problem, review optimal solutions and understand the intuition. Group similar problems to recognize patterns. Practice explaining your approach out loud. Review problems after a few days to test retention.

What is the optimal approach for "Split the Array to Make Coprime Products"?

For this hard-level Array problem, the key is to identify the core pattern. Analyze the input constraints to determine acceptable time complexity. Consider whether hash table techniques can optimize the solution. Always handle edge cases and validate your approach with examples before coding.

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