Introduction

In this article, we delve into finding the sum of the greatest four-digit even square number using the J programming language. The process involves breaking down the problem using logical steps and mathematical insights, which can be applied to other similar problems involving even squares and digit sums.

Understanding Even Squares

Firstly, let's understand what an even square number is. An even square number is one that results from squaring an even number. For instance, 982 9604 is an even square number because 98 is even.

The Greatest Four-Digit Even Square

The task at hand is to find the sum of the greatest four-digit even square number. To start, we need to identify the greatest four-digit even square number. The greatest four-digit number is 9999, but since it is not even, we need to consider the nearest even number, which is 9998.

Brute Force Approach Using J Programming Language

Using the J programming language, we can find the greatest four-digit even square number by checking the squares of even numbers from the range 90 to 99:

./n~.n.ev 9000 to 9999

This J code snippet iterates through the even numbers from 9000 to 9999, finds the largest even square number, and prints it. The output of this code is 9604.

Sum of the Digits

Step-by-Step Approach

To find the sum of the digits of 9604, we break the number down:

9   6   0   4  19

Thus, the sum of the digits of 9604 is 19.

Alternative Calculation Method

Another method to find the greatest four-digit even square is to consider the square of 98:

982 (100 - 2)2 10000 - 400 4 9604

This method simplifies the calculation and confirms the correct result.

General Formula for Even Squares in Different Bases

The calculations above can be generalized to understand the sum of the digits for even square numbers in different bases. For bases where the square of the largest four-digit number is an even square, we can use the following formulas:

In any even number base (b), the sum of the digits of the largest four-digit even square is given by (2b - 1) In binary (base 2), there are no four-digit even squares In base 4, the largest four-digit even square is (32_4^2 3010_4), and the sum of the digits is 4 In odd bases (b), the largest four-digit even square is given by (b^2 - 1^2), and the sum of the digits is (2b - 2)

These formulas provide a concise way to calculate the sum of the digits for even squares in various bases, making it a useful tool for further exploration.

Conclusion

Understanding the greatest four-digit even square and the sum of its digits can be achieved through both specific examples and general formulas. The J programming language offers a convenient tool for brute force checking, while the provided formulas generalize the solution for different bases. This method of solving the problem can be applied to other similar problems, enhancing our mathematical reasoning and problem-solving skills.