Rational Numbers, Finite Cyclic Monoids, Divisibility Rules, and Numbers of Type 99 … 900 … 0

Abstract There is a curious connection between decimal representations of rational numbers, the structure of finite cyclic monoids, divisibility rules between integers, and divisors of the numbers of the form 99 … 900 … 0. In all of these cases, we find not only periodicity from some point on, but also the same type of periodicity.