An interesting fact about the number 123456789 and its generalization in arbitrary base

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,

$$\begin{eqnarray} 123456789&*2=& 246913578\\ &*4=& 493827156\\ &*5=& 617283945\\ &*7=& 864197523\\ &*8=& 987654312 \end{eqnarray}$$