Counting 5-Digit Perfect Squares Divisible by 5: An In-Depth Guide

In this comprehensive guide, we will explore how to count 5-digit perfect squares that are multiples of 5. We will use both mathematical reasoning and programming techniques to solve this problem. By the end of this article, you will understand the logic behind the solution and how to implement it programmatically.

Introduction to Perfect Squares and Multiples of 5

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 15625 is a 5-digit perfect square because its square root is 125, which is an integer. In this context, we are specifically interested in 5-digit perfect squares that are also multiples of 5. A number is a multiple of 5 if it ends in 0 or 5.

Mathematical Approach

Let's start by identifying the smallest and largest 5-digit perfect squares. The square root of 10 is approximately 3.16, so the smallest 5-digit perfect square is the square of 100, which is 10000. The square root of 99999 is approximately 316, so the largest 5-digit perfect square is the square of 316, which is 99856. Therefore, any number between 100 and 316, inclusive, will yield a 5-digit perfect square when squared.

Now, for a number to be a multiple of 5, it must end in 0 or 5. We need to count how many numbers between 100 and 316 have squares ending in 0 or 5. The range of numbers to consider is from 100 to 316 inclusive, which is a total of 217 numbers. However, not all of these numbers will result in a 5-digit perfect square that is a multiple of 5.

First, we will calculate the total number of 5-digit perfect squares. This is simply the difference between 316 and 100, which is 217. However, not all of these perfect squares are multiples of 5. To find the number of multiples of 5, we can use the formula:

[ text{Number of multiples of 5} leftlfloor frac{217-1}{5} rightrfloor 1 44 ]

This formula accounts for the fact that we start counting from 100, which is a multiple of 5.

Brute Force Approach Using J Programming Language

To verify the solution using a brute force approach, we can use the J programming language. The J code to find 5-digit perfect squares that are multiples of 5 is as follows:

m . n~: : n . ~0  510000 to 9999944

The output of this code confirms that there are 44 such numbers.

The list of 5-digit perfect squares that are multiples of 5 can be printed using the following J code:

m10000 11025 12100 13225 14400 15625 16900 18225 19600 21025 22500 24025 25600 27225 28900 30625 32400 34225 36100 38025 40000 42025 44100 46225 48400 50625 52900 55225 57600 60025 62500 65025 67600 70225 72900 75625 78400 81225 84100 87025 90000 93025 96100 99225

Python Implementation

We can also solve this problem using Python. The following Python code will list all 5-digit perfect squares that are multiples of 5:

print "List of 5-digit perfect squares which are multiples of 5:"cnt  0for k in range(10000, 100000):    for n in range(1, int(k**0.5) 1):        if n*n  k and k % 5  0:            print k            cnt   1print "There are", cnt, "5-digit perfect squares which are multiples of 5."

The output of this code is as follows:

List of 5-digit perfect squares which are multiples of 5:10000 11025 12100 13225 14400 15625 16900 18225 19600 21025 22500 24025 25600 27225 28900 30625 32400 34225 36100 38025 40000 42025 44100 46225 48400 50625 52900 55225 57600 60025 62500 65025 67600 70225 72900 75625 78400 81225 84100 87025 90000 93025 96100 99225There are 44 5-digit perfect squares which are multiples of 5.

This confirmation using Python further verifies our solution.

We hope this guide helps you understand the process of counting and listing 5-digit perfect squares that are multiples of 5. It combines both mathematical reasoning and programming skills to provide a comprehensive solution to this problem.

Conclusion

By using the mathematical approach and verifying it with both J programming language and Python, we have determined that the number of 5-digit perfect squares divisible by 5 is 44. This article should serve as an excellent resource for anyone looking to understand and solve similar problems.

Good luck with your own explorations in mathematics and programming!