The terminating decimals can all be written as fractions whose denominators are powers of ten. So if a fraction in simplest form has a denominator with a prime factor other than 2 or 5, the decimal repeats.
@Tennessee Sure, given a fraction in lowest terms, whenever the denominator of the fraction contains a factor that is relatively prime (AKA coprime) with the base, there will be repeats. But all the technical language makes it harder to follow.
