Check If It Is a Good Array

Problem Description

Given an array nums of positive integers. Your task is to select some subset of nums, multiply each element by an integer and add all these numbers. The array is said to be good if you can obtain a sum of 1 from the array by any possible subset and multiplicand.

Return True if the array is good otherwise return False.

Key Insights

• The problem can be reduced to determining if the greatest common divisor (GCD) of the array is 1.
• If the GCD of the array is 1, it implies that there exist integer coefficients that can combine the elements to form any integer, including 1.
• The Euclidean algorithm can be used to compute the GCD efficiently.
• The solution must handle large arrays and large integer values efficiently due to constraints.

Space and Time Complexity

Time Complexity: O(n) - where n is the number of elements in nums, since we compute the GCD in a linear pass. Space Complexity: O(1) - only a constant amount of space is used.

Solution

To solve the problem, we will compute the GCD of the entire array. If the GCD is 1, we can form the sum 1 using the integers in the array with appropriate coefficients. We will utilize the Euclidean algorithm to find the GCD of two numbers iteratively for the entire array.

Code Solutions