Integer which is Multiplied by 9 when moving Last Digit to First/Mistake

Source Work

 * The Dictionary
 * $10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$
 * $10,112,359,550,561,797,752,808,988,764,044,943,820,224,719$

Mistake

 * The unique number which, when the last digit, $9$, is moved to the front, is the number multiplied by $9$.

Correction
The number is not actually unique.

Let $N$ be the number given.

Then the positive integers formed by concatenating the decimal representation of $N$ with itself any number of times have the same property:
 * $\sqbrk {NN}, \sqbrk {NNN}, \sqbrk {NNNN}, \ldots$