 A Mathematician at play| Children

# A tower of exponents and the power of divisibility Many students learn divisibility rules in school. These can be fun to play with, but they are often a little tame, and they won’t help you much here. Today I’d like to play around with some stranger, more surprising divisibility puzzles.

## THE POWER OF 37

Let’s play a quick game. Now follow these steps: 1) Pick any digit. 2) Write this digit three time in a row to get a 3-digit number. 3) Divide this three digit number by the same digit added to itself three times. For example, if I picked 5, I would get 37 as my answer.

555 ÷ (5 + 5 + 5) = 37

Now here’s the question: Can you get any number aside from 37 as your answer? Why or why not?

## THE 7, 11, 13 MYSTERY

We are going to be playing another game now. It’s similar to the one in the first puzzle, and yet, it is dierent. Here are the steps: 1) Choose a three digit number. 2) Write it twice in a row to get a 6-digit number. That’s it. We are good to go now.

For our example, I choose 356, and write it twice to get 356,356.

I claim this number is divisible by 7. Indeed, 356,356 ÷ 7 = 50908

I claim this quotient is divisible by 11. Indeed, 50908 ÷ 11 = 4628

I claim this quotient is disable by 13. Indeed, 4628 ÷ 13 = 356

And look! The final quotient is the original number!

Tackle this question: Can you find a three digit number where this won’t happen? Or will it happen for every three-digit number. And if so, why?

## NUMBERS THAT BREAK CALCULATORS

One fun thing about exponents is that they get big fast, and this is especially true of towers of exponents.

To calculate a tower of exponents, start from the top and work your way down. For example,

So 4^3^2 = 4 ^ 9 = 262,144.

Meanwhile, 5^4^3 is forty-five digits long.

Now that you know how towers of exponents work, here’s the question: Given below are two towers of exponents being added to each other. Can you tell me if the sum of those two huge numbers divisible by 11? 