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

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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 by hypothesis:

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

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 $a \ne 0$ by hypothesis:

$F > 0 \cdotp 1$

Hence:

$z = 9$

So:

Decimal Expansion

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

$\Box$

Hence the value for $N$.

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