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?

One logic behind.
$9 \times (0 + 2) = 9 \times 0 + 9 \times 2 = 18$
$9 \times (10 + 2) = 9 \times 10 + 9 \times 2 = 108$
$9 \times (110 + 2) = 9 \times 110 + 9 \times 2 = 1008$
So on.
