Power of Base minus 1 is Repdigit Base minus 1

Theorem

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

Let $n = b^k - 1$ for some integer $k$ such that $k \ge 1$.

Let $n$ be expressed in base $b$ representation.

Then $n$ is a repdigit number consisting of $k$ instances of digit $b - 1$.

$\ds \dfrac {b^k - 1} {b - 1}$

$=$

$\ds \sum_{j \mathop = 0}^{k - 1} b^j$

$\ds \leadsto \ \ $

$\ds n = b^k - 1$

$=$

$\ds \paren {b - 1} \sum_{j \mathop = 0}^{k - 1} b^j$

$\ds $

$=$

$\ds \sum_{j \mathop = 0}^{k - 1} \paren {b - 1} b^j$

Thus, by the definition of base $b$ representation, $n$ consists of $k$ occurrences of the digit $b - 1$.

$\blacksquare$