Cyclic decimal divisibility
= Cyclic decimal divisibility
For an $L$-digit decimal integer $n$, moving its leading digit to the end gives a cyclic rotation $c_1(n)$ satisfying
$$
c_1(n)\equiv10n\pmod{10^L-1}.
$$
Thus, when a modulus $d$ divides $10^L-1$, every rotation has the same divisibility by $d$ as $n$, up to multiplication by a power of ten.