What is the optimal time complexity of solving Minimum operations to make GCD multiple of k?

The optimized solution achieves a time complexity of O(n) by using sorting, mathematical shortcuts, or hash table lookups.

What is the auxiliary space complexity for Minimum operations to make GCD multiple of k?

The space complexity is O(n) since we only use a minimal/constant amount of extra memory for tracking variables (or auxiliary space if dynamic structures are allocated).

What are the typical edge cases we must handle in Minimum operations to make GCD multiple of k?

Key edge cases include empty collections, negative or large bounds causing integer overflow, arrays with only one element, or inputs where target criteria are not met.

How can we optimize the brute force approach for Minimum operations to make GCD multiple of k?

Brute force methods often inspect all possible combinations. We optimize this by sorting the input list to enable binary search, or tracking processed items in a set to avoid redundant nested loop lookups.

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Minimum operations to make GCD multiple of k

Learn how to solve the 'Minimum operations to make GCD multiple of k' problem. This detailed resource details brute force and optimized approaches.

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Problem Statement

Easy

Write a function min_operations_gcd_k(arr, k) that returns the minimum number of operations to make the Greatest Common Divisor (GCD) of all elements in the array a multiple of k. In one operation, you can either increment or decrement any element of the array by 1.

Constraints

•1 <= len(arr) <= 10^5
•1 <= k <= 10^4
•1 <= arr[i] <= 10^9

Examples

Example 1

Input

min_operations_gcd_k([4, 5, 6], 5)

Output

Explanation

Increment 4 to 5 (1 op) and decrement 6 to 5 (1 op). The array becomes [5, 5, 5] whose GCD is 5, which is a multiple of 5.

Example 2

Input

min_operations_gcd_k([2, 3], 3)

Output

Explanation

Increment 2 to 3 (1 op). The array becomes [3, 3] whose GCD is 3, which is a multiple of 3.

Need a Hint?

Analyze the input constraints. Try sorting first (O(n log n)) or using a hash map/set to track seen elements in O(n) time.

Edge Cases to Watch

Empty list or null input variables
Single item lists/arrays
Extremely large input bounds causing integer or stack overflow

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