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

Corollary to Integer which is Multiplied by 9 when moving Last Digit to First
Let $N$ be the positive integer:
 * $N = 10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$

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

Proof
Let's say $N$ is written as $abc...xyz$ ($a\neq 0$). Define a repeating decimal number $F=0.abc…xyzabc…xyzabc…xyz…$. We then know that $9F=0.zabc…xyzabc…xyzabc…xy…$, and therefore $90F=z.abc…xyzabc…xyzabc…xyz…$. Subtracting, $90F-F=$ $89F=$ $z$, so $F=z/89$. Since $F$ must be greater than 0.1 (because $a$ isn't zero), we need $z=9$. Using an infinite precision calculator (or working by hand!) we find $F=0.10112359550561797752808988764044943820224719…$ and we get the value for $N$; you can take the repeating part once, twice, thrice, etc., and they all solve the problem.