More general - what is the hard problem of recovering r from r*p mod q? I would like to know the cryptographic hard problem that is most closely tied to recovering integer $r$ from the modular product $r\times p\mod q$.  (This is a simplification of an earlier post that had some errors).  It really looks like integer factorization to me; if not, what else would it be?

More specifically, select two prime numbers $p$ and $q$, $q&gt;p$, and a random positive integer $r$, large enough such that $q/p&lt;r&lt;q$.  Publish $q$, but keep $p$ and $r$ private.  Further, assume there are several instances of $r$ for a given pair of $\langle p, q\rangle$ to work with.  Assuming existence of a hardness problem X, such that a polynomial-time solution of X could be reduced to finding either $r$ or $p$ from the integer $$r\times p\mod q$$in polynomial time, what is this problem X? 

I am relatively new to this.  I looked at a few hard problems; none of the residuosity or discrete logarithmic problems seem to apply, but I'm hesitant to say that it's integer factorization or RSA in case there is some problem with a stronger assumption that fits.  I want to get a good characterization of the construct so that I may describe it accurately.

Thanks for your help and patience!
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.