Question

1250. Check If It Is a Good Array

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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.

 

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

Solutions

Solution 1: Mathematics (Bézout's Identity)

First, consider the situation where we select two numbers. If the selected numbers are $a$ and $b$, then according to the problem's requirements, we need to satisfy $a \times x + b \times y = 1$, where $x$ and $y$ are any integers.

According to Bézout's Identity, if $a$ and $b$ are coprime, then the above equation definitely has a solution. In fact, Bézout's Identity can also be extended to the case of multiple numbers. That is, if $a_1, a_2, \cdots, a_i$ are coprime, then $a_1 \times x_1 + a_2 \times x_2 + \cdots + a_i \times x_i = 1$ definitely has a solution, where $x_1, x_2, \cdots, x_i$ are any integers.

Therefore, we only need to determine whether there exist $i$ coprime numbers in the array nums. The necessary and sufficient condition for two numbers to be coprime is that their greatest common divisor is $1$. If there exist $i$ coprime numbers in the array nums, then the greatest common divisor of all numbers in the array nums is also $1$.

So we transform the problem into: determining whether the greatest common divisor of all numbers in the array nums is $1$. We can do this by traversing the array nums and finding the greatest common divisor of all numbers in the array nums.

The time complexity is $O(n + \log m)$, and the space complexity is $O(1)$. Where $n$ is the length of the array nums, and $m$ is the maximum value in the array nums.

class Solution:
  def isGoodArray(self, nums: List[int]) -> bool:
    return reduce(gcd, nums) == 1

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);
  }
}

class Solution {
public:
  bool isGoodArray(vector<int>& nums) {
    int g = 0;
    for (int x : nums) {
      g = gcd(x, g);
    }
    return g == 1;
  }
};

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)
}