Represent K^N as The Sum of Exactly N numbers in Java

Representing K^N as the sum of exactly N numbers in Java requires careful consideration of mathematical principles and programming techniques.

Problem Statement

We have given two integers N (exponent integer) and K (base integer). We have to represent KN as the sum of exactly N numbers. Print N/A if no such numbers are possible.

Examples

Input: N = 5, K = 2

Output: 2 2 4 8 16

Approach: In order to obtain numbers such that their sum is a power of K, we can choose the numbers that follow the condition:

i^th number = Kⁱ-k^i-1

It gives the sum as a power of K.

Let's check the above equation through an example.

Let N = 5 and K = 2.

We need to represent 25 (=32) as the sum of exactly 5 numbers

According to the mentioned approach, the 5 numbers which can be chosen are

(2¹) = 2

(2² - 2¹) = 4 - 2 = 2

(2³ - 2²) = 8 - 4 = 4

(2⁴ - 2³) = 16 - 8 = 8

(2⁵ - 2⁴) = 32 - 16 = 16

Adding the numbers = 2 + 2 + 4 + 8+ 16 = 32 which is clearly 2⁵

Therefore, the required 5 numbers are 2, 2, 4, 8, and 16.

Approach and Methodology

To achieve this, several methods can be explored based on the constraints and requirements of the problem. We will cover two primary methods: a straightforward adjustment approach and a more balanced distribution approach.

Simple Distribution and Adjustment Approach

Step-by-Step Implementation

1. Calculate K^N

Use the Math.pow() function to compute K^N. Since we are dealing with integers, cast the result to int.

2. Initialization

Create an array numbers of size 𝑁 to store the 𝑁 integers.

3. Distribute and Adjust

Initialize all elements of numbers to 1 initially.
Adjust the first element to ensure the sum equals K^N. Specifically, set numbers[0]= K^N - (N - 1).

File Name: PowerSumMethods.java

 
public class PowerSumMethods {
    // Method 1: Simple Distribution and Adjustment
    public static int[] representAsSumMethod1(int K, int N) {
        // Calculate K^N
        int power = (int) Math.pow(K, N);       
        // Create an array to store N numbers that sum up to K^N
        int[] numbers = new int[N];    
        // Initialize all elements to 1 initially
        for (int i = 0; i < N; i++) {
            numbers[i] = 1;
        }        
        // Adjust the first element to make the sum correct
        numbers[0] = power - (N - 1);      
        return numbers;
    }
    public static void main(String[] args) {
        int K = 3; // Base number
        int N = 4; // Exponent
        // Get the representation using Method 1
        int[] numbers = representAsSumMethod1(K, N);
        // Print the numbers
        System.out.print("Method 1 Numbers: ");
        for (int num : numbers) {
            System.out.print(num + " ");
        }
        System.out.println();
        // Verify the sum
        int sum = 0;
        for (int num : numbers) {
            sum += num;
        }
        System.out.println("Method 1 Sum: " + sum);
    }
}   

Output:

 
Method 1 Numbers: 78 1 1 1 
Method 1 Sum: 81

Explanation:

Calculation: K^N is computed using Math.pow(K, N) and cast to int.

Initialization: An array numbers of size N is initialized with each element set to 1.

Adjustment: The first element of numbers is adjusted to ensure the sum equals K^N.

Verification: The sum of elements in numbers is verified to ensure correctness.

Balanced Distribution Approach

Step-by-Step Implementation

Calculate K^N:

Compute K^N using Math.pow() and cast to int.

Determine Base and Remainder

Divide K^N by 𝑁 to determine a base value.
Calculate the remainder to distribute it among the numbers.

Distribution

Initialize each element of numbers with the base value.
Distribute the remainder across the array to ensure the sum equals K^N.

File Name: PowerSumMethods.java

 
 public class PowerSumMethods {
    // Method 2: Balanced Distribution
    public static int[] representAsSumMethod2(int K, int N) {
        // Calculate K^N
        int power = (int) Math.pow(K, N);   
        // Create an array to store N numbers that sum up to K^N
        int[] numbers = new int[N];        
        // Determine base and remainder
        int base = power / N;
        int remainder = power % N;        
        // Fill the array with base values
        for (int i = 0; i < N; i++) {
            numbers[i] = base;
        }       
        // Distribute the remainder
        for (int i = 0; i < remainder; i++) {
            numbers[i]++;
        }        
        return numbers;
    }
    public static void main(String[] args) {
        int K = 3; // Base number
        int N = 4; // Exponent
        // Get the representation using Method 2
        int[] numbers = representAsSumMethod2(K, N);
        // Print the numbers
        System.out.print("Method 2 Numbers: ");
        for (int num : numbers) {
            System.out.print(num + " ");
        }
        System.out.println();
        // Verify the sum
        int sum = 0;
        for (int num : numbers) {
            sum += num;
        }
        System.out.println("Method 2 Sum: " + sum);
    }
}  

Output:

 
Method 2 Numbers: 21 20 20 20 
Method 2 Sum: 81

Explanation

Calculation: K^N is computed using Math.pow(K, N) and cast to int.

Initialization: An array numbers of size N is initialized with each element initially set to a base value.

Distribution: The remainder of K^N divided by N is distributed across numbers to ensure the sum equals K^N.

Verification: The sum of elements in numbers is verified to ensure correctness.

Selecting the Approach

Simple Adjustment is straightforward and effective for smaller values of K and 𝑁.

Balanced Distribution provides a more evenly distributed representation, especially when K and N are larger.

Complexity

Time Complexity: O(N)

Space Complexity: O(1)

Conclusion

In Java, representing K^N as the sum of exactly N numbers involves both mathematical computation and programming techniques. Depending on the scenario and requirements, you can choose between a simple adjustment approach or a more balanced distribution method. These methods ensure accuracy and efficiency in computing the desired representation of K^N as a sum of N integers.

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