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.
Example 1:
Input: nums = [12,5,7,23] Output: true Explanation: Pick numbers 5 and 7. 5*3 + 7*(-2) = 1
Example 2:
Input: nums = [29,6,10] Output: true Explanation: Pick numbers 29, 6 and 10. 29*1 + 6*(-3) + 10*(-1) = 1
Example 3:
Input: nums = [3,6] Output: false
Constraints:
1 <= nums.length <= 10^51 <= nums[i] <= 10^9This problem is rooted in number theory and revolves around understanding how integers can combine to form other numbers. A key mathematical insight is that if a set of integers can generate the value 1 through integer linear combinations, the array is considered "good". This idea is closely related to properties of the greatest common divisor (GCD).
Instead of trying all combinations of numbers (which would be computationally infeasible), an efficient strategy is to analyze the collective GCD of the entire array. By iteratively computing the GCD of elements, you can determine whether the array has the necessary property to produce 1. This dramatically reduces the problem from exponential exploration to a simple linear pass through the array.
The optimized approach processes each element once while maintaining a running gcd. This results in O(n log M) time complexity (where M is the maximum value in the array) and O(1) additional space.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative GCD across the array | O(n log M) | O(1) |
Ashish Pratap Singh
Use these hints if you're stuck. Try solving on your own first.
Eq. ax+by=1 has solution x, y if gcd(a,b) = 1.
Can you generalize the formula?. Check Bézout's lemma.
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The GCD determines the smallest positive integer that can be formed through integer linear combinations of the array elements. If the GCD of all numbers is 1, Bézout's identity guarantees that 1 can be constructed using those numbers, making the array 'good'.
Problems involving GCD properties and number theory occasionally appear in technical interviews at large tech companies. While this exact problem may not always appear, similar questions testing mathematical insight and efficient computation are common in FAANG-style interviews.
No special data structure is required for this problem. A simple iteration over the array while maintaining a running GCD value is sufficient. This keeps the space usage constant and the implementation straightforward.
The optimal approach relies on number theory, specifically the properties of the greatest common divisor (GCD). By computing the GCD of all elements in the array, you can determine whether the numbers can form the value 1 through integer combinations. If the overall GCD equals 1, the array satisfies the required condition.