There are a lot of good suggestions in this feed, but here are a few problems that let you introduce modular arithmetic.
First, one can easily prove that an integer mod 9 is equal to the sum of its digits mod 9.
Second, you can prove Fermat's little theorem k^p mod p = k where p is prime.
I suppose that even (a+b) mod n = (a mod n + b mod n) mod n is kind of neat too.
You can prove that the calendar repeats itself every 28 years.