Numbers That Stay True To Themselves

“Do you know numbers that stay true to themselves?” I asked my son.

“What do you mean?”

“Let me give you an example. The number 25, if you multiply it by itself, what’s the result?”

“625.”

“The last two digits of 625 is 25. It is where we started! The number 25 stays true to itself after multiplying itself!”

“A-Ha! Interesting!”

I continued, “We call 25 an automorphic number. It’s sometimes also called a curious number. When an automorphic number multiplies itself, the product ends with the original number. Let’s start with single digit numbers. Can you tell me which single digit numbers are automorphic?”

Son started going through the numbers, and quickly concluded 0, 1, 5, 6 are single-digit automorphic numbers.

0 * 0 = 0

1 * 1 = 1

5 * 5 = 25

6 * 6 = 36

“What about two-digit numbers?”

After quite some calculations, son replied, “25 and 76.”

25 * 25 = 625

76 * 76 = 5776

“For 3-digit numbers, there are still two automorphic numbers: 625, 376

625 * 625 = 390625

376 * 376 = 141376

Now can you tell me if there is any pattern here?”

Son contemplated, “Well, there seem to be 2 automorphic numbers for 1-digit, 2-digit, 

3-digit numbers each, except 0 and 1.”

“Right, 0 and 1 are kind of special, so we can ignore them in this case.”

“And, 5, 25, 625, they just keep growing!”

“Nice, the last digits of an automorphic number are automorphic numbers themselves! What else?”

“Hmm, I can’t think of more.”

“Let me give you some hints.

    5 + 6 = 11

  25 + 76 = 101

625 + 376 = 1001”

“Ha, so the sum of the two 4-digit automorphic numbers is 10001!”

“That’s right! So next time, when somebody tells you to stay true to yourself, you can say sure, that’s a good idea. And I know some numbers that are staying true to themselves!”