Let $x$ be a positive integer of the form $nnn\frac n2 \frac n2 \frac n2$ where $n$ is an even integer less than $30$.
Then $\frac x \pi$ is of the form $abab0c.\ ...$ where $a$ and $b$ are digits between $0$ and $9$, and $c$ is either $0$ or $1$. If $n>8$ carry digits as shown below.
For example, if $n=12$, then $x$ would be $(12)(12)(12)666=(12)(13)2666=(13)32666=1332666$, and $\frac x \pi=424200.76...$.
Proof by exhaustion is easy in this case as there is only a short list to test for. But why does this phenomenon occur? (it doesn't work for $3$, $3.14$ or even $3.141$ as divisors). Is there a shorter and more elegant proof to this?