For all of those in 405, (math for teachers), I was reading our book and discovered that one of the proofs of our theorem's is based on Euler's phi function! So the theorem is "Suppose that m/n is in lowest terms and that m < n. Then m/n has a simple-periodic decimal representation if and only if 2 or 5 are not factors of n." Since we know that every rational m/n that is represented by a simple-periodic decimal is equal to a fraction with a denominator consisting of all 9's, the proof used Euler's Theorem to provide the denominator of 9s.

a theorem based on Euler's theorem