The problem is described as follows. Let $c_1=p_1q_1+r$, $c_2=p_2q_2+r$, $\cdots$, $c_n=p_nq_n+r$, where $p_i$'s, $q_i$'s, $r$ are all large positive integers, and $p_i$'s and $q_i$'s are randomly chosen. Notice that $p_i$'s and $q_i$'s are different while $r$ is the same. Now one knows $c_1,\cdots,c_n$, and he knows the above structure of $c_1,\cdots,c_n$, but he does not know $p_i$'s or $q_i$'s. He aims to extract the value of $r$ from $c_1,\cdots,c_n$. Is this problem hard (either proven to be hard, or no polynomial solving algorithm found yet)? Does this problem have a name?
A similar (but easier) problem is solving the system of linear congruences: by knowing $c_1,\cdots,c_n$ and $p_1,\cdots,p_n$, solve $r$ from the equations $r\equiv c_1\mod p_1$, $r\equiv c_2\mod p_2$, $\cdots$, $r\equiv c_n\mod p_n$. This problem is related to the Chinese remainder theorem and I know that it has polynomial-time solving algorithms. However, in my question in the first paragraph above, one only knows $c_1,\cdots,c_n$, but does not know the moduli $p_1,\cdots,p_n$ (or $q_1,\cdots,q_n$), the problem of my question might be harder. I can only find works on linear congruences but cannot find related works on my question. Is anyone familiar with this question? I just wonder is this a problem that has been ever studied.