Divisors of Repunit with Composite Index

Theorem
Let $R_n$ be a repunit number with $n$ digits.

Let $n$ be composite such that $n = r s$ where $1 < r < n$ and $1 < s < n$.

Then $R_r$ and $R_s$ are both divisors of $R_n$.

Proof
Let $n = r s$.

Then:

Similarly:
 * $R_n = R_s \paren {\ds \sum_{j \mathop = 0}^{r - 1} 10^{s j} }$

Thus, for example:

The pattern is clear.