I'm trying to go through various bits of neat, fun math with some junior-high-school students in my local area, and am thinking of doing a short unit on modular arithmetic/finite groups.
I'm looking for some fun number facts that can be shown using modular arithmetic. For example, the fact that the sum of three cubes can't be a multiple of 9 +/- 4.
Any such fun tidbits people could point me to would be very much appreciated.
On
My main use for modular arithmetic at the age was for this:
http://everything2.com/title/Cracking+a+Master+Lock
It's interesting and practical (e.g. when you forget a combination for an old lock), though naturally judgment should be exercised before volunteering this kind of information to Jr. High students.
You probably know this, but the fact that the remainder of a number modulo $9$ is the same as that of the sum of its digits is a simple calculation involving powers of $10 = 1$. There's a similar formula modulo $11$, where $10 = -1$.
Also, the fact that if $2^n - 1$ is prime (a Mersenne prime), then $n$ is prime. If $n = ab$, then reduce $2^n - 1$ modulo $2^a - 1$, where you have $2^a = 1$.