Numbers the Multiple of whose Reciprocal are Cyclic Permutations/Examples/49

Example of Numbers the Multiple of whose Reciprocal are Cyclic Permutations
Let $n \in \Z$ such that $1 \le n < 49$ and that $7 \nmid n$, where $\nmid$ denotes that $7$ is not a divisor of $n$.

The digits in the decimal expansion of the rational number $\dfrac n {49}$, for $1 \le n < 49$, form a cyclic permutation of the digits in the decimal expansion of $\dfrac 1 {49}$:

Patterns abound.