Is there any Information on the "Modular Approximate Greatest Divisor" problem?
Specifically: is the problem of obtaining $\ p\ $ from samples of $\ pq_i + e_i \bmod x$ with secret $\ p, q_i, e_i$ and public $x$ equivalent in hardness to the previously mentioned variant of the problem that operates over integers? 


Actually, if $p$ is relatively prime to $x$, and the $q_i$ values are uniformly distributed over $[0, x-1]$, then it is strictly harder than the AGCD problem, in that it is informationally secure (rather than computationally secure), that is, even if we assume that attacker has unbounded computational capabilities, he still cannot recover $p$.

The demonstration is actually fairly simple; even if we assume that attacker knows $e_i$, then for any sample value $s = pq_i + e_i \bmod x$, for any potential value $p'$, there exists a unique $q'_i$ value, namely, $p'^{-1}(s - e_i) \bmod x$ that would yield the observed sample $s$, hence the attacker gains no information about the relative probability of any candidate value $p$ (except for whether it's relatively prime to $x$)