Structure of Recurring Decimal

Theorem
Let $\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits with no nonperiodic part.

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

Then $\dfrac 1 {m n}$ has a nonperiodic part of $q$ digits, and a recurring part of $p$ digits.