Distinct Prime Factors of Product of Array

Problem Description

Given an array of positive integers nums, return the number of distinct prime factors in the product of the elements of nums.

Key Insights

• A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself.
• The product of the elements can be large; however, we only need to count the distinct prime factors, not calculate the product explicitly.
• Factorization can be efficiently performed since the maximum value of elements in nums is 1000, which allows precomputation of prime factors.

Space and Time Complexity

Time Complexity: O(n * log(m)), where n is the number of elements in nums and m is the maximum value of elements in nums (in this case, 1000).
Space Complexity: O(k), where k is the number of distinct prime factors.

Solution

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

1. Precompute the prime factors for all integers from 2 to 1000 using a modified Sieve of Eratosthenes.
2. For each number in the input array nums, retrieve its prime factors and store them in a set to ensure uniqueness.
3. Finally, return the size of the set, which represents the number of distinct prime factors.

Code Solutions