014-construct-minimum-sum-array

Construct Minimum Sum Array

Problem Description

Little C wants to construct an array of n elements that satisfies the following conditions:

1. All elements in the array are pairwise distinct
2. The greatest common divisor (GCD) of all elements is k
3. The sum of the array elements should be as small as possible

Output the minimum possible sum of the array elements that satisfies the conditions.

Examples

Example 1

Input:

n = 3, k = 1

Output:

6

Explanation: The array [1,2,3] can be constructed, with a sum of 6

Example 2

Input:

n = 2, k = 2

Output:

6

Explanation: The array [2,4] can be constructed, with a sum of 6

Example 3

Input:

n = 4, k = 3

Output:

30

Explanation: The array [3,6,9,12] can be constructed, with a sum of 30

Solution Approach

1. Understanding the requirements:

   • We need to find n distinct numbers
   • The GCD of these numbers must be k
   • The sum of these numbers should be as small as possible

2. Key observation:

   • For the GCD to be k, all numbers must be multiples of k
   • To minimize the sum, we should choose the smallest multiples of k
   • Simply select consecutive multiples of k starting from k

3. Algorithm steps:

   • Initialize an empty array result
   • Starting from k, add one multiple of k at a time
   • Continue until the array length reaches n
   • Return the sum of the array elements

Implementation

def solution(n: int, k: int) -> int:
    # Create an array to store the selected numbers
    result = []
    # Starting from k, find n multiples of k
    current = k
    while len(result) < n:
        result.append(current)
        current += k

    # Return the sum of array elements
    return sum(result)

Code Explanation

1. result = []: Create an empty array to store the selected numbers

2. current = k: Start from k, since k is the smallest possible value

3. while len(result) < n:: Loop until n numbers are found

   • result.append(current): Add the current number to the array
   • current += k: Move to the next multiple of k

4. return sum(result): Return the sum of all elements in the array

Complexity Analysis

• Time Complexity: O(n), as we need to find n numbers
• Space Complexity: O(n), as we need to store an array of n numbers

Walkthrough Example

Taking n=4, k=3 as an example:

1. Initial current=3, result=[3]
2. current=6, result=[3,6]
3. current=9, result=[3,6,9]
4. current=12, result=[3,6,9,12]
5. Return 3+6+9+12=30

Summary

1. Do not try to find non-multiples of k, as that cannot guarantee the GCD is k
2. Do not skip multiples of k, as that would result in a non-minimal sum
3. Do not forget the condition that array elements must be pairwise distinct

Related problems:

• GCD-related problems
• Constructive algorithm problems
• Array minimum sum problems