Question

Look at several examples of rational numbers in the form $\frac{p}{q}(q\ne 0)$, where $p$ and $q$ are integers with no common factors other than $1$ and having terminating decimal representaions (expansions). Can you guess what property $q$ must satisfy?

Answer 1

The property that $q$ must satisfy in order that the rational numbers in the

from $\frac{p}{q}$ , where $p$ and $q$ are integers with no

common factor other than $1$, have maintaining decimal representation is

prime factorization of $q$ has only powers of $2$ or power of $5$ or both .

i.e $2^m\times 5^n$ , where $m=1,2,3,\cdots$ or $n=1,2,3,\cdots$