## Fractions with Terminating Decimals

Late last month, we have talked about fractions with terminating decimals as well fractions with non-terminating decimals. We ended up with a conjecture that a fraction is a terminating decimal if its denominator has only the following factors: 2 (or its powers), 5 (or its powers) or both.  In this post,we refine this conjecture. This conjecture is the same as saying

A rational fraction $\frac{a}{b}$ in the lowest terms has a terminating decimal if and only if the integer $b$ has no prime factor other than $2$ and $5$.

Note that we have already explained the only if part in the preceding post. It remains to show that if part which is

if $\frac{a}{b}$ is in lowest terms and $b$ contains at most $2$ and $5$ as factors, then the fraction is a terminating decimal. » Read more