4 of 4 try 3 at stating the problem. It's definitely several values of r for a given and fixed p, q.

# 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 $$q$$, $$q>p$$, and a random positive integer $$r$$, large enough such that $$q/p. Publish $$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!