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 $9 N$.

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 $N$ be written in the form:
 * $\sqbrk {abc \ldots xyz}$

where $a \ne 0$.

Let us define the recurring decimal:
 * $F = \sqbrk {0 \cdotp \dot abc \ldots xy \dot z}$

Hence :
 * $9 F = \sqbrk {0 \cdotp \dot zabc \ldots x \dot y}$

Therefore:
 * $90 F = \sqbrk {z \cdotp abc \ldots xy \dot z}$

Hence by subtraction:


 * $90 F - F = 89 F = z$

so:
 * $F = \dfrac z {89}$

Because $a \ne 0$ :
 * $F > 0 \cdotp 1$

Hence:
 * $z = 9$

So:

Decimal Expansion
Hence the value for $N$.

We can take the recurring part an arbitrary number of times, and they all solve the problem.