# Kaprekar … one more time!

Following up on my previous post titled “A surprising result …” involving Kaprekar’s Operation, I thought I would offer one more interesting result from the Indian mathematician D. R. Kaprekar.  So let’s talk about a Kaprekar Number.

What’s a Kaprekar number?  Take an n-digit number k. Square it and add the right n digits to the left n or n-1 digits. If the resultant sum is k, then k is called a Kaprekar number.  Oh, is that all?  I thought it was going to be something complicated to understand!

Let me show you by example.

• Take the number 45.  This means that k = 45.  Since it is a 2 digit number we have n = 2.
• Square it.  Now we have 2025.  The right 2 digits are 25 and the left 2 digits are 20.  (Remember n = 2.)
• If you look at the sum of 20 and 25 you get 45.
• So we call 45 a Kaprekar number.

Let’s try a second example.

• Take the number 297.  This means that  k = 297.  Since it is a 3 digit number we have n = 3.
• Square it.  Now we have 88209.  The right 3 digits are 209 and left 2 digits (remember n or n-1) are 88.
• If you look at the sum of 209 and 88 you get 297.
• So we call 297 a Kaprekar number.

Amazing, isn’t it?  Can you think of any other Kaprekar Numbers?  There are plenty of them out there in the great mathematical universe.

If you are interested in reading more about the theory behind Kaprekar Numbers, you can look at the paper titled “The Kaprekar Numbers” by Douglas E. Iannucci published in the Journal of Integer Sequences.

(Thanks to MathWorld for the official definition of a Kaprekar Number:  Weisstein, Eric W. “Kaprekar Number.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/KaprekarNumber.html)