Length of Reciprocal of Product of Powers of 2 and 5

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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 $q$.


Proof


Sources