Structure of Recurring Decimal

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Theorem

Let $\dfrac 1 m$, when expressed as a decimal expansion, recur with a period of $p$ digits.

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.


Proof


Sources