Multiple of Repdigit Base minus 1

Theorem
Let $b \in \Z_{>1}$ be an integer greater than $1$.

Let $n$ be a repdigit number of $k$ instances of the digit $b - 1$ for some integer $k$ such that $k \ge 1$.

Let $m \in \Z_{>1}$ be an integer such that $1 < m < b$.

Then $m \times n$, when expressed in base $b$, is of the form:


 * $n = \left[{r d d \cdots d s}\right]_b$

where:
 * $d = b - 1$
 * $r = m - 1$
 * $s = b - m$
 * there are $k - 1$ occurrences of $d$.

Proof
which is exactly the representation $\left[{r d d \cdots d s}\right]_b$ as defined.