495 has an almost magical property and is known as the Kaprekar constant for 3 digit numbers. What does that mean?

Take any 3 digit number that has at least 2 different digits. Write the digits from greatest to least to create a new 3-digit number. From that number subtract the same digits written in reverse order. Repeat this process, and you will get the number 495 in no more than seven iterations. This graphic shows this process applied to the number 101.

The following puzzle can easily transform into a multiplication table if you first find all the square factors of the given clues. Go ahead give it a try!

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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- 495 is a composite number.
- Prime factorization: 495 = 3 x 3 x 5 x 11, which can be written 495 = (3^2) x 5 x 11
- The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 495 has exactly 12 factors.
- Factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495
- Factor pairs: 495 = 1 x 495, 3 x 165, 5 x 99, 9 x 55, 11 x 45, or 15 x 33
- Taking the factor pair with the largest square number factor, we get √495 = (√9)(√55) = 3√55 ≈ 22.248595

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I love and hate Kaprekar’s numbers at the same time. That’s because when I was still in high school, I “discovered” this property on 6174 and I really thought that I have made an awesome discovery, but it turned out that someone had already found this property a few decades earlier than me haha…

I also know that feeling except when I’ve “discovered” something, I found out someone else knew about it centuries before. Missing it by a few decades has to be much more disappointing. The rediscovery process can still be fun and enlightening.

Yes, it’s quite disappointing, though it’s also quite fun for me to rediscover already known theorems. This happens to me a lot in number theory.

Intriguing. Is there a simple reason for this?

I read that it took Kaprekar three years to prove that 6174 has the same property so I don’t know that a simple reason exists. However, note that the difference of any two numbers with the same digits will always be divisible by 9 after the first and subsequent iterations. Both 495 and 6174 are divisible by 9 and yield themselves.