Ezvor

3867. Sum of GCD of Formed Pairs

MediumArray Preparing judge… LeetCode

You are given an integer array nums of length n.

Construct an array prefixGcd where for each index i:

  • Let mxi = max(nums[0], nums[1], ..., nums[i]).
  • prefixGcd[i] = gcd(nums[i], mxi).

After constructing prefixGcd:

  • Sort prefixGcd in non-decreasing order.
  • Form pairs by taking the smallest unpaired element and the largest unpaired element.
  • Repeat this process until no more pairs can be formed.
  • For each formed pair, compute the gcd of the two elements.
  • If n is odd, the middle element in the prefixGcd array remains unpaired and should be ignored.

Return an integer denoting the sum of the GCD values of all formed pairs.

The term gcd(a, b) denotes the greatest common divisor of a and b.

 

Example 1:

Input: nums = [2,6,4]

Output: 2

Explanation:

Construct prefixGcd:

i nums[i] mxi prefixGcd[i]
0 2 2 2
1 6 6 6
2 4 6 2

prefixGcd = [2, 6, 2]. After sorting, it forms [2, 2, 6].

Pair the smallest and largest elements: gcd(2, 6) = 2. The remaining middle element 2 is ignored. Thus, the sum is 2.

Example 2:

Input: nums = [3,6,2,8]

Output: 5

Explanation:

Construct prefixGcd:

i nums[i] mxi prefixGcd[i]
0 3 3 3
1 6 6 6
2 2 6 2
3 8 8 8

prefixGcd = [3, 6, 2, 8]. After sorting, it forms [2, 3, 6, 8].

Form pairs: gcd(2, 8) = 2 and gcd(3, 6) = 3. Thus, the sum is 2 + 3 = 5.

 

Constraints:

  • 1 <= n == nums.length <= 105
  • 1 <= nums[i] <= 10​​​​​​​9
ArrayMathTwo PointersSortingSimulationNumber Theory

Hints

Hint 1
Maintain a running prefix maximum mxi while iterating nums to compute prefixGcd[i] = gcd(nums[i], mxi).
Hint 2
Sort prefixGcd in non-decreasing order.
Hint 3
Form pairs by combining smallest unpaired and largest unpaired elements.
Hint 4
Compute gcd for each pair and sum them; ignore middle element if n is odd.