 Solution

LetBy Fermat's little theorem, $l\mid p-1$. If the digit count $L$ is a multiple of $l$, then $10^L\equiv 1(\mathrm{mod}p)$. The cyclic decimal divisibility relation givesand hence also modulo $p$. Since $p>7$ is coprime to $10$, every $10^j$ is a unit modulo $p$. Thereforefor every $j$. Since $c_0(n)=n$, this proves the required equivalence.