Length of Reciprocal of Product of Powers of 2 and 5

Theorem

Let $n \in \Z$ be an integer.

Let $\dfrac 1 n$, when expressed as a decimal expansion, terminate after $m$ digits.

Then $n$ is of the form $2^p 5^q$, where $m$ is the greater of $p$ and $q$.

Since $\dfrac 1 n$ terminates after $m$ digits:

$\dfrac {10^m} n$ is an integer

$\dfrac {10^{m - 1}} n$ is not an integer

From the first condition, we have $n = 2^p 5^q$ for some positive integers $p, q \le m$.

This gives $m \ge \max \set {p, q}$.

From the second condition, we cannot have both $p, q \le m - 1$.

Therefore at least one of $p, q$ is equal to $m$.

This gives $m \le \max \set {p, q}$.

These results give $m = \max \set {p, q}$.

$\blacksquare$