Revolving Number

Remember when you first learned the decimal representation of $\displaystyle\frac{1}{k}$? Some of them have a nice form $\left( \dfrac{1}{2} = 0.5, \dfrac{1}{4} = 0.25, \dfrac{1}{5} = 0.2 \right)$, others don't terminate but the pattern is easy to remember $\left( \dfrac{1}{3} = 0.333\cdots \right)$, and the remaining ones have the form you might not care about. Among those which lie on the final category, this one particularly attracted me

\[ \frac{1}{7} = 0.\overline{142857} \]

and led my train of thought to an even bigger mystery.

The repeated value $142857$ is special on its own way. Try jotting down the first multiples of the number and you'll see why
\begin{align*} 142857 \cdot 2 & = 285714 \\
142857 \cdot 3 & = 428571 \\
142857 \cdot 4 & = 571428 \\
142857 \cdot 5 & = 714285 \\
142857 \cdot 6 & = 857142 \\
\end{align*}
The pattern is obvious: these consecutive multiples of $142857$ can be obtained by rotating the original number as if the final digit connected to the first digit (imagine the digits are sitting on a circular table). I'd like to define numbers like these as 'revolving numbers' due to their nature. A more formal definition is the following,

A circular representation of $N \in \mathbb{N}$ is the placement of its digits in a circle without changing the order. $M \in \mathbb{N}$ is a rotation of $N$ if the circular representation of $M$ can be obtained from the circular representation of $N$. Finally, $N$ is a revolving number if the first $k>1$ multiple of $N$ is a rotation of $N$ and the number of digits of $N$ equals $k$.

An initial observation from the definition of a revolving number $N$ is that its number of digits must be less than $10$; since if $k = 10$, then $10 \cdot N$ would have more digits than $N$. Hence, $N < 10^9$ and we can easily find all revolving numbers with a simple brute-force program. The challenge is for us to do it without using the aid of computer.

Apparently, the list is extremely thin within the first one billion positive integers. Let's introduce a more generous definition where $k$ is not necessarily as big as the number of digits,

$N$ is a partially-revolving number if for some $k>1$ less or equal to the number of digit of $N$, the first $k$ multiple of $N$ is a rotation of $N$ .

In other words, if $2N$ is a rotation of $N$, then $N$ is a partially-revolving number. Seems there are much more of these, but can you find one on top of your head aside from $142857$?

Surprisingly, $142857$ is the only revolving number. The title of this blog is actually a hint, since I left the noun as singular. And the more amazing fact is that the other partially-revolving numbers are only its multiples: $285714$ and $428571$.