Maximum Subarray With Equal Products

Problem Description

You are given an array of positive integers nums. An array arr is called product equivalent if prod(arr) == lcm(arr) * gcd(arr), where prod(arr) is the product of all elements of arr, gcd(arr) is the GCD of all elements of arr, and lcm(arr) is the LCM of all elements of arr. Return the length of the longest product equivalent subarray of nums.

Key Insights

• The relationship between product, GCD, and LCM can be used to determine if a subarray is product equivalent.
• The GCD of a subarray can be determined using the Euclidean algorithm, while the LCM can be calculated using the relationship lcm(a, b) = abs(a*b) / gcd(a, b).
• A sliding window approach can be useful to explore all potential subarrays efficiently.
• Since the constraints are small, we can afford to check multiple subarrays in a brute-force manner with optimizations.

Space and Time Complexity

Time Complexity: O(n^2) - in the worst case, we check each subarray. Space Complexity: O(1) - we use a constant amount of extra space.

Solution

The solution involves iterating through all possible subarrays of the input array nums. For each subarray, we calculate the product, GCD, and LCM of the elements. If the condition prod(arr) == lcm(arr) * gcd(arr) holds true, we keep track of the length of that subarray. The maximum length found during this process is returned as the result.

1. Use two nested loops to explore all subarrays.
2. For each subarray:
   • Calculate the product of elements.
   • Calculate the GCD of elements using a helper function.
   • Calculate the LCM based on the GCD.
   • Check if the product equals the product of LCM and GCD.
3. Update the maximum length whenever a product equivalent subarray is found.

Code Solutions