Powers of 10 Expressible as Product of 2 Zero-Free Factors

Theorem
The powers of $10$ which can be expressed as the product of $2$ factors neither of which has a zero in its decimal representation are:

Proof
Let $p q = 10^n$ for some $n \in \Z_{>0}$.

Then $p q = 2^n 5^n$.

, suppose $p = 2^r 5^s$ for $r, s \ge 1$.

Then $2 \times 5 = 10$ is a divisor of $p$ and so $p$ ends with a zero.

Thus for $p$ and $q$ to be zero-free, it must be the case that $p = 2^n$ and $q = 5^n$ (or the other way around).

The result follows from Powers of 2 and 5 without Zeroes.