For each integer $n\ge 3$, let $a_n$ denote the base-$n$ number $0.{\overline{12}}_n$. The product $a_5{a}_6\dots a_{100}$ can be expressed as $\frac{m}{n!}$, where $m$ and $n$ are positive integers and $n$ is as small as possible. What is the value of $m$?