The number $142,857$ is widely known as a cyclic number, meaning consecutive multiples are cyclic permutations, i.e.
$1 × 142,857 = 142,857$
$2 × 142,857 = 285,714$
$3 × 142,857 = 428,571$
and so on.
142857 is the repeating unit of $\frac{1}{7} = 0.\overline{142857}$ and in fact, every prime for which $10$ is a primitive root will generate a cyclic number (if we allow $0$ as a first digit, for example $0588235294117647$ which is the repeating unit of $\frac{1}{17}$). These primes are called full reptend primes.
From what I've read, it seems that there is a bijection between full reptend primes and cyclic numbers: a number is cyclic if and only if it is the repeating unit for the reciprocal of a full reptend prime.
I was able to find a proof for the if part. I was wondering if anyone can provide a proof for the only if part.
Edit: It's been a while since I posed this question and I still haven't found a proof. I've added a bounty in hopes of prompting some interest.