Is this a good math trick?

I have been reading Arthur Benjamin’s book Magic of Maths to get some ideas on how to present math and to also learn a few new things. There is a really good chapter on Fibonacci series, which shows some interesting ways of demonstrating relationships among Fibonacci numbers.

One of the things that the book does is to use math tricks to introduce certain ideas. This inspired me to come up with this way to show an example of a rather significant mathematical result.

Using a calculator, enter an odd number less than 50. Hit the x^2 button. Hit it again. Then divide by 8. The decimal portion of the number will always be .125.

Thinking about this, it should be clear that hitting the x^2 key twice is the same as raising a number to the fourth power. The .125 decimal just means that there is a remainder of 1 when you divide by 8. You get the number 4 by counting all the numbers less than 8 that do not share a divisor with 8, which would just be the 4 numbers 1, 3, 5 and 7.

The generalization, due to the great mathematician Leonhard Euler, is that if you take any number n and count the k numbers less than n that do not have a common factor then if you take any number a that does not share a factor with n, a^k will always have a remainder of 1 when divided by n.

This may or may not get your juices flowing, but it is considered an important mathematical discovery and has been used for practical purposes as part of data encryption.