The number $(12\ldots(b-1))$ in base $b$ has the property that when multiplied by any integer $1\le k\le b-1$ which is coprime to $b-1$, its digits are permuted. Why?
For example in base 10,
\begin{eqnarray}
123456789&*2=& 246913578\\
&*4=& 493827156\\
&*5=& 617283945\\
&*7=& 864197523\\
&*8=& 987654312
\end{eqnarray}
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