Why is $9 \times 11{\dots}12 = 100{\dots}08$? While I was working on Luhn algorithm implementation, I discovered something unusual.
$$ 9 \times 2 = 18 $$
$$ 9 \times 12 = 108 $$
$$ 9 \times 112 = 1008 $$
$$ 9 \times 1112 = 10008 $$
Hope you can observe the pattern here.
What to prove this?
What is it's significance?

The repunit, $R_k = \overbrace{111\ldots 111}^{k \text{ ones}}$ , can be written as $R_k = \dfrac{10^k-1}{9}$
Your nice pattern corresponds to $9\times (R_k+1) = (10^k-1)+9 = 10^k+8$
