More general - what is the hard problem of recovering r from r*p mod q?
When given a single triple consisting of $(p,q,x)$ with $x = r \cdot p \mod q$, then there is no hard problem. It takes one inversion and one multiplication (both in modular arithmetic) to calculate $r$.

If just $x$ is given, then you can choose $p$ and $q$ arbitrarily and calculate a matching $r$ to fullfill $x = rp \mod q$.

If the actual question is about recovering the original values: That's not possible. This is exactly the same situation like only giving some bounds for random values (e.g. $q&gt;x$) and nothing else.