Characterising reals with terminating decimal expansions

This is a consequence of FTA = Fundamental Theorem of Arithmetic (existence and uniqueness of prime factorizations of integers).

Suppose the real $\rm\,r\,$ has terminating decimal expansion with $\rm\:k\:$ digits $\neq 0$ after the decimal point. Then multiplying it by $\rm\,10^k$ shifts the decimal point right by $\rm\,k\,$ digits, hence yields an integer, i.e. $\rm\: 10^k r = n\in \Bbb Z.\:$ Thus $\rm\, r = n/10^k\,$ so canceling common factors to reduce this fraction to lowest terms yields a fraction whose denominator divides $\rm\:10^k\! = 2^k 5^k.\:$ By unique factorization the only such divisors are $\rm\:2^i 5^j\:$ for $\rm\:i,j \le k.\:$ Also by unique factorization the lowest terms representation of a fraction is unique, so there cannot exist another equivalent fraction in lowest terms whose denominator has prime factors other than $2$ and $5$. This completes the proof.

Exactly the same argument works if we replace $10$ by any other radix.