Find the Maximum Factor Score of Array

Find the Maximum Factor Score of Array

What is this problem about?

The Find the Maximum Factor Score of Array coding problem defines the "factor score" of an array as the product of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of all its elements. You are allowed to remove at most one element from the array to maximize this score. Your task is to return the maximum achievable factor score.

Why is this asked in interviews?

This problem tests your knowledge of Number Theory interview patterns, specifically GCD and LCM properties. It evaluation your ability to handle multiple scenarios (removing each element vs. removing none) and your proficiency with large-number arithmetic. Companies like Info Edge use this to see if you can implement mathematical identities efficiently.

Algorithmic pattern used

This is a Brute Force with Number Theory problem.

1. Implement helper functions for gcd(a, b) and lcm(a, b).
2. Calculate the factor score for the entire array.
3. Iterate through the array and, for each index $i$, calculate the GCD and LCM of the array excluding the element at $i$.
4. Maintain a global maximum of these scores.
5. Optimization: To avoid $O(N^2)$ recalculations, you can use Prefix and Suffix arrays to store the running GCD and LCM from both ends of the array. This allows you to find the GCD/LCM of the array minus index $i$ in $O(1)$ time after $O(N)$ preprocessing.

Example explanation

nums = [2, 4, 8]

1. Full array: GCD=2, LCM=8. Score = 16.
2. Remove 2: [4, 8]. GCD=4, LCM=8. Score = 32.
3. Remove 4: [2, 8]. GCD=2, LCM=8. Score = 16.
4. Remove 8: [2, 4]. GCD=2, LCM=4. Score = 8. Max Score: 32.

Common mistakes candidates make

• Overflow: Factor scores (especially LCM) can grow extremely large, exceeding 64-bit integers.
• LCM property: Forgetting that $LCM(a, b, c) = LCM(LCM(a, b), c)$.
• Complexity: Using a naive $O(N^2)$ approach for large arrays instead of prefix/suffix optimization.

Interview preparation tip

Always remember the relationship: $LCM(a, b) = (a imes b) / GCD(a, b)$. When calculating the LCM of an entire array, update the running LCM step-by-step to prevent overflow as much as possible, though the final score may still require large integer support depending on the language.