Smallest Number which is Multiplied by 99 by Appending 1 to Each End

Theorem
The smallest positive integer which is multiplied by $99$ when $1$ is appended to each end is:
 * $112 \, 359 \, 550 \, 561 \, 797 \, 732 \, 809$

Proof
We have that:


 * $112 \, 359 \, 550 \, 561 \, 797 \, 732 \, 809 = 7 \times 11 \times 61 \times 87 \, 629 \times 337 \, 411 \times 809 \, 063$

while:

Let $N$ be the smallest integer satisfying $99 N = \sqbrk {1N1}$ when expressed in decimal notation.

Suppose $N$ is $k$ digits long.

Then:
 * $\sqbrk {1 N 1} = 10^{k + 1} + 10 N + 1$

Subtracting $10 N$ from $99 N$ gives:
 * $89 N = 10^{k + 1} + 1$

One can show, by trial and error, that the smallest $k$ where $10^{k + 1} + 1$ is divisible by $89$ is $21$.

Then $N = \dfrac {10^{22} + 1} {89} = 112 \, 359 \, 550 \, 561 \, 797 \, 732 \, 809$.