Why is $9 \times 11{\dots}12 = 100{\dots}08$?

Question

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?

It's so good to see when people get interested in these beautiful things! and they start looking for explanation :) — Daniel, Jan 21 '17 at 14:07
Try doing the multiplications with pencil and paper rather than relying on a calculator ... — hmakholm left over Monica, Jan 21 '17 at 14:45
Hint $\ 9 = 10!-!1\ $ divides $\ 10^N!+!8 = 10^N!-!1 + (10-1)\ \ $ See Casting Out Nines. — Bill Dubuque, Jan 21 '17 at 15:26

score 7 · Answer 1 · answered Jan 21 '17 at 14:10

7

Here's an idea.

This pattern is easier to understand: $$ 9 \times 11 = 99$$

$$9 \times 111 = 999$$

$$9 \times 1111 = 9999$$

Then see what happens when you add 9 to both sides.

answered Jan 21 '17 at 14:10

Pat Devlin

score 5 · Accepted Answer · answered Jan 21 '17 at 14:16

5

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 $9\times (R_k+1) = (10^k-1)+9 = 10^k+8$

answered Jan 21 '17 at 14:16

Joffan

score 3 · Answer 3 · answered Jan 21 '17 at 14:14

3

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.

answered Jan 21 '17 at 14:14

Kanwaljit Singh

score 0 · Answer 4 · answered Jan 21 '17 at 14:11

0

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

answered Jan 21 '17 at 14:11

B. Goddard

score 0 · Answer 5 · answered Jan 21 '17 at 14:13

0

Write $1...12$ as $$\overline{1...(\times\ n)...1} +1=\frac{10^n-1}{9}+1$$

Hence $$9\times (\frac{10^n-1}{9}+1)=10^n-1+9=10^n+8$$

answered Jan 21 '17 at 14:13

8hantanu

5 Answers5