Condition for Repunits to be Coprime

Theorem
Let $R_p$ and $R_q$ be repunit numbers with $p$ and $q$ digits respectively.

Then $R_p$ and $R_q$ are coprime $p$ and $q$ are coprime.

Necessary Condition
Let $R_p$ and $R_q$ be coprime.

$p$ and $q$ are not coprime.

Let $p = d m, q = d n$ for some $d > 1$.

Then by Divisors of Repunit with Composite Index:
 * $R_d \mathrel \backslash R_p$

and:
 * $R_d \mathrel \backslash R_q$

where $\backslash$ denotes divisibility.

That is, $R_d$ is a common divisor of $R_p$ and $R_q$.

This contradicts our hypothesis that $R_p$ and $R_q$ are coprime.

Thus by Proof by Contradiction, $p$ and $q$ are coprime.