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?

$$9 \times 11\cdots12 = 9 \times (11\cdots 11+1) = 99\cdots99 + 9.$$
