Number of Different Subsequences GCDs

Problem Description

You are given an array nums that consists of positive integers. The task is to return the number of different GCDs among all non-empty subsequences of nums.

Key Insights

• A subsequence can be formed by removing some elements of the array, and it is crucial to note that all non-empty subsequences must be considered.
• The GCD of a set of numbers is the greatest integer that divides all the numbers in the set without leaving a remainder.
• The maximum possible GCD for any subsequence is limited by the largest number in the input array.
• We can leverage the properties of GCD and the relationship between numbers to count distinct GCD values efficiently.

Space and Time Complexity

Time Complexity: O(n log(max(nums))) - where n is the size of the input array and max(nums) is the maximum value in the array. Space Complexity: O(max(nums)) - for storing counts of potential GCDs.

Solution

To solve this problem, we can use the following approach:

1. Create a boolean array to track which GCDs are possible from the numbers in the input array.
2. Iterate through each integer from 1 to the maximum number in the array, and for each integer, check if it can be formed as a GCD of any subsequence.
3. For each integer, check if it can divide any number in the input array. If it can divide at least one number, it is a candidate GCD.
4. Count all the unique GCD values found in this process.

Code Solutions