How can we prove that every rational number has a terminating or periodic decimal form?

Question

The title says it all. I'm aware of the proof of the converse of my statement, but how do I go on about proving this. Any help would be appreciated.

Answer 1

When dividing through n, you can only have n possible remainders, from $0$ to $n-1$, so either the division stops, when one of these remainders is $0$, or loops, when one of these non-zero remainders is equal to a previous non-zero remainder.