Difference between Kaprekar Number and Square is Multiple of Repunit

Theorem
Let $n$ be a Kaprekar number of $m$ digits.

Then:
 * $n^2 - n = k R_m$

where:
 * $R_m$ is the $m$-digit repunit
 * $k$ is an integer.

Proof
Since $n$ is a Kaprekar number of $m$ digits:
 * $\begin {cases} n^2 = a \times 10^m + b \\ n = a + b \end {cases}$

for some positive integers $a$ and $b$.

Hence: