Exploring 3-Digit Perfect Cubes Whose Digit Sum is a Perfect Square

In this article, we delve into the intriguing world of 3-digit perfect cubes. Specifically, we aim to identify how many of these numbers have a digit sum that constitutes a perfect square. Perfect squares are numbers that result from squaring integers. We will follow a systematic approach to explore this question, providing a detailed breakdown of each step involved.

Identifying the Range of 3-Digit Perfect Cubes

First, we need to determine which 3-digit numbers fall within the range of perfect cubes. The smallest 3-digit number is 100, and the largest is 999.

The Smallest Integer

The smallest integer ( n ) such that ( n^3 geq 100 ) is 5, since ( 5^3 125 ).

The Largest Integer

Similarly, the largest integer ( n ) such that ( n^3 leq 999 ) is 9, since ( 9^3 729 ).

The Relevant Integers

Therefore, the integers ( n ) we are interested in are 5, 6, 7, 8, and 9. Let's calculate their cubes:

( 5^3 125 ) ( 6^3 216 ) ( 7^3 343 ) ( 8^3 512 ) ( 9^3 729 )

Thus, the 3-digit perfect cubes are: 125, 216, 343, 512, and 729.

Calculating the Sum of Digits for Each Cube

Next, we calculate the sum of the digits for each of these cubes:

( 125 ): The sum of the digits is ( 1 2 5 8 ). ( 216 ): The sum of the digits is ( 2 1 6 9 ). ( 343 ): The sum of the digits is ( 3 4 3 10 ). ( 512 ): The sum of the digits is ( 5 1 2 8 ). ( 729 ): The sum of the digits is ( 7 2 9 18 ).

Checking Which Sums are Perfect Squares

We need to identify which of these sums are perfect squares less than or equal to 18. The perfect squares in this range are 0, 1, 4, 9, and 16. Let's check each sum:

( 8 ): Not a perfect square. ( 9 ): Perfect square, since ( 9 3^2 ). ( 10 ): Not a perfect square. ( 8 ): Not a perfect square. ( 18 ): Not a perfect square.

Only the sum of the digits for 216, which is 9, is a perfect square.

Conclusion

Thus, there is only one 3-digit perfect cube whose sum of digits is a perfect square: 216.

Python Code Implementation

Here is a Python code snippet to automate the process and check the sums of the digits for 3-digit perfect cubes:

from math import sqrt () def is_perfect_square(n): return int(sqrt(n))**2 n def sum_of_digits(n): return sum(int(digit) for digit in str(n)) def perfect_cubes_with_perfect_square_sum(): for n in range(5, 10): cube n**3 digit_sum sum_of_digits(cube) if is_perfect_square(digit_sum): print(f"Cube: {cube}, Sum of Digits: {digit_sum}") perfect_cubes_with_perfect_square_sum()

This code defines a function to check if a number is a perfect square and another function to calculate the sum of the digits of a number. It then iterates through the relevant integers, calculates their cubes, finds the digit sums, and checks if the sums are perfect squares. The result confirms that only 216 has a perfect square sum of digits.