Given two sets of positive integers $\{a_i\}$ and $\{b_j\}$, $N \nmid a_i$ and $M \nmid b_j$ for any $i$ and $j$.
I would like to know if it is possible to derive a possible value of $P$ from $M$ and $N$ directly, where $P \nmid (a_i+b_j)$ for any combinations of $i$ and $j$.