LeetCode #1250 — HARD

Check If It Is a Good Array

Break down a hard problem into reliable checkpoints, edge-case handling, and complexity trade-offs.

The Problem

Problem Statement

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 <= 10^5
1 <= nums[i] <= 10^9

Step 01

Brute Force Baseline

Problem summary: 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.

Baseline thinking

Start with the most direct exhaustive search. That gives a correctness anchor before optimizing.

Pattern signal: Array · Math

Example 1

[12,5,7,23]

Example 2

[29,6,10]

Example 3

[3,6]

Step 02

Core Insight

What unlocks the optimal approach

Eq. ax+by=1 has solution x, y if gcd(a,b) = 1.
Can you generalize the formula?. Check Bézout's lemma.

Interview move: turn each hint into an invariant you can check after every iteration/recursion step.

Step 03

Algorithm Walkthrough

Iteration Checklist

Define state (indices, window, stack, map, DP cell, or recursion frame).
Apply one transition step and update the invariant.
Record answer candidate when condition is met.
Continue until all input is consumed.

Use the first example testcase as your mental trace to verify each transition.

Step 04

Edge Cases

Minimum Input

Single element / shortest valid input

Validate boundary behavior before entering the main loop or recursion.

Duplicates & Repeats

Repeated values / repeated states

Decide whether duplicates should be merged, skipped, or counted explicitly.

Extreme Constraints

Largest constraint values

Re-check complexity target against constraints to avoid time-limit issues.

Invalid / Corner Shape

Empty collections, zeros, or disconnected structures

Handle special-case structure before the core algorithm path.

Step 05

Full Annotated Code

Source-backed implementations are provided below for direct study and interview prep.

// Accepted solution for LeetCode #1250: Check If It Is a Good Array
class Solution {
    public boolean isGoodArray(int[] nums) {
        int g = 0;
        for (int x : nums) {
            g = gcd(x, g);
        }
        return g == 1;
    }

    private int gcd(int a, int b) {
        return b == 0 ? a : gcd(b, a % b);
    }
}

// Accepted solution for LeetCode #1250: Check If It Is a Good Array
func isGoodArray(nums []int) bool {
	g := 0
	for _, x := range nums {
		g = gcd(x, g)
	}
	return g == 1
}

func gcd(a, b int) int {
	if b == 0 {
		return a
	}
	return gcd(b, a%b)
}

# Accepted solution for LeetCode #1250: Check If It Is a Good Array
class Solution:
    def isGoodArray(self, nums: List[int]) -> bool:
        return reduce(gcd, nums) == 1

// Accepted solution for LeetCode #1250: Check If It Is a Good Array
struct Solution;

impl Solution {
    fn is_good_array(nums: Vec<i32>) -> bool {
        let mut res = nums[0];
        let n = nums.len();
        for i in 0..n {
            res = gcd(res, nums[i]);
            if res == 1 {
                return true;
            }
        }
        false
    }
}

fn gcd(mut m: i32, mut n: i32) -> i32 {
    while m != 0 {
        let temp = m;
        m = n % temp;
        n = temp;
    }
    n.abs()
}

#[test]
fn test() {
    let nums = vec![12, 5, 7, 23];
    let res = true;
    assert_eq!(Solution::is_good_array(nums), res);
    let nums = vec![29, 6, 10];
    let res = true;
    assert_eq!(Solution::is_good_array(nums), res);
    let nums = vec![3, 6];
    let res = false;
    assert_eq!(Solution::is_good_array(nums), res);
}

// Accepted solution for LeetCode #1250: Check If It Is a Good Array
function isGoodArray(nums: number[]): boolean {
    return nums.reduce(gcd) === 1;
}

function gcd(a: number, b: number): number {
    return b === 0 ? a : gcd(b, a % b);
}

Step 06

Interactive Study Demo

Use this to step through a reusable interview workflow for this problem.

Custom checkpoints (one per line)

Press Step or Run All to begin.

Step 07

Complexity Analysis

Time

O(n + log m)

Space

O(1)

Approach Breakdown

BRUTE FORCE

O(n²) time

O(1) space

Two nested loops check every pair or subarray. The outer loop fixes a starting point, the inner loop extends or searches. For n elements this gives up to n²/2 operations. No extra space, but the quadratic time is prohibitive for large inputs.

OPTIMIZED

O(n) time

O(1) space

Most array problems have an O(n²) brute force (nested loops) and an O(n) optimal (single pass with clever state tracking). The key is identifying what information to maintain as you scan: a running max, a prefix sum, a hash map of seen values, or two pointers.

Shortcut: If you are using nested loops on an array, there is almost always an O(n) solution. Look for the right auxiliary state.

Coach Notes

Common Mistakes

Review these before coding to avoid predictable interview regressions.

Off-by-one on range boundaries

Wrong move: Loop endpoints miss first/last candidate.

Usually fails on: Fails on minimal arrays and exact-boundary answers.

Fix: Re-derive loops from inclusive/exclusive ranges before coding.

Overflow in intermediate arithmetic

Wrong move: Temporary multiplications exceed integer bounds.

Usually fails on: Large inputs wrap around unexpectedly.

Fix: Use wider types, modular arithmetic, or rearranged operations.