【leetcode】1250. Check If It Is a Good Array

Topics are as follows:

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

Problem-solving ideas: See this subject, we might be able to guess this question corresponds to mathematical laws. As to what the law is, I do not know. Later, the Internet search found a corresponding law is Shu Pei theorem, the final solution is to find the common denominator of all the elements, you can determine whether 1.

Shu Pei Theorem (or Bezu Theorem , Bézout's identity) is named after the French mathematician Etienne Pei Shu, it describes any integer , b, and their largest convention a

The number of d, on the unknowns of Indeterminate Equation x and y (referred to as PEI Shu equation): if a, b are integers and gcd (a, b) = d , then for any integers x, y, ax + by must be a multiple of d are, in particular, certain integers x, y, ax + by = d so established.

It is an important corollary: a, b coprime necessary and sufficient condition is the presence of integers x, y so ax + by = 1.

code show as below:

class Solution(object):
    def isGoodArray(self, nums):
        """
        :type nums: List[int]
        :rtype: bool
        """
        def gcd(m, n):
            if not n:
                return m
            else:
                return gcd(n, m % n)
        val = nums[0]
        for i in range(1,len(nums)):
            val = gcd(val,nums[i])
        return val == 1