Fermat's Little Theorem

A neat application of Fermat's Little Theorem is repeating decimals. It seems pretty intuitive that the decimal expansion of any rational number should eventually repeat itself, if it does not terminate. But there is no limit to how long the repeating segment can be, and it could take ages to know for sure that the decimal repeats through standard division.

Can you use Fermat's Little Theorem to prove this fact in the special case where the denominator of the rational number is prime? (A generalization of FLT can prove this for any denominator.)

Write down your proof, then look at the solution when you are ready. Obviously your proof may be somewhat different from the one shown here.