2709 - Greatest Common Divisor Traversal.

Hard

You are given a 0-indexed integer array nums, and you are allowed to traverse between its indices. You can traverse between index i and index j, i != j, if and only if gcd(nums[i], nums[j]) > 1, where gcd is the greatest common divisor.

Your task is to determine if for every pair of indices i and j in nums, where i < j, there exists a sequence of traversals that can take us from i to j.

Return true if it is possible to traverse between all such pairs of indices, or false otherwise.

Example 1:

Input: nums = 2,3,6

Output: true

Explanation: In this example, there are 3 possible pairs of indices: (0, 1), (0, 2), and (1, 2). To go from index 0 to index 1, we can use the sequence of traversals 0 -> 2 -> 1, where we move from index 0 to index 2 because gcd(nums0, nums2) = gcd(2, 6) = 2 > 1, and then move from index 2 to index 1 because gcd(nums2, nums1) = gcd(6, 3) = 3 > 1. To go from index 0 to index 2, we can just go directly because gcd(nums0, nums2) = gcd(2, 6) = 2 > 1. Likewise, to go from index 1 to index 2, we can just go directly because gcd(nums1, nums2) = gcd(3, 6) = 3 > 1.

Example 2:

Input: nums = 3,9,5

Output: false

Explanation: No sequence of traversals can take us from index 0 to index 2 in this example. So, we return false.

Example 3:

Input: nums = 4,3,12,8

Output: true

Explanation: There are 6 possible pairs of indices to traverse between: (0, 1), (0, 2), (0, 3), (1, 2), (1, 3), and (2, 3). A valid sequence of traversals exists for each pair, so we return true.

Constraints:

• <code>1 <= nums.length <= 10⁵</code>

• <code>1 <= numsi<= 10⁵</code>