More generally, the question of why a decimal expansion repeats can now be seen as the question of why repeats every P-1, when P is prime. For a prime number, P, every number less than that number is coprime to it, so ϕ(P)=P-1. As long as 10 is coprime to the denominator, this generates numbers that are also coprime to the denominator. So , the set of numbers (mod M) coprime to M, form a group under multiplication. This leads us to consider the group of which is the multiplication mod M group of numbers coprime to M (the not-coprime-case will be considered in a damn minute).

When you want to outsmart the world, you turn to the facts. And the facts are in the science.