Show that if $x$ has a terminating decimal expansion then $x=p/q$ for integers $p,q$ where the only prime factors of q are 2's and 5's.

Question

I've proven the converse statement but don't know where to start for this statement. Would you suggest doing a proof by contradiction?

Answer 1

If $x=0.a_1a_2\ldots a_k$, then \begin{align*} x& =0.a_1a_2\ldots a_k\\ & =\frac{a_1a_2\ldots a_k}{10^k} \end{align*}

The denominator has only possible prime factors as $2$ and $5$.