Numbers the Multiple of whose Reciprocal are Cyclic Permutations

Theorem
Let $m \in \Z_{>0}$.

Consider the reciprocal of $m$.

Let $n \in \Z$ such that $1 \le n < m$.

Then:
 * The digits in the decimal expansion of the rational number $\dfrac n m$ form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 m$


 * $(1): \quad m$ is the integer power of a prime number $p$
 * $(2): \quad$ The period of recurrence of the decimal expansion of $1 / p$ is $p - 1$
 * $(3): \quad p$ is not a divisor of $n$.
 * $(3): \quad p$ is not a divisor of $n$.