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

Theorem
Let $N$ be the positive integer:
 * $N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$

$N$ is the smallest positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$.

Also see

 * Integer which is Multiplied by Last Digit when moving Last Digit to First: the example given here is a special case