Integer which is Multiplied by 9 when moving Last Digit to First/Proof 2

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 $9 N$.

Proof

From Integer which is Multiplied by Last Digit when moving Last Digit to First, $N$ is equal to the recurring part of the fraction:

$q = \dfrac {a_1} {10 a_1 - 1}$

where $a_1 = 9$.

Thus:

$q = \dfrac 9 {10 \times 9 - 1} = \dfrac 9 {89}$

Hence:

Decimal Expansion

$\dfrac 9 {89} = 0 \cdotp \dot 10112 \, 35955 \, 05617 \, 97752 \, 80898 \, 87640 \, 44943 \, 82022 \, 471 \dot 9$

$\Box$

The result follows.

$\blacksquare$