Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following $100$ $9$-digit integers:

\[
\begin{align*}
700 \cdot 712 \cdot 718 + 320,\\
701 \cdot 713 \cdot 719 + 320,\\
702 \cdot 714 \cdot 720 + 320,\\
&\vdots\\
798 \cdot 810 \cdot 816 + 320,\\
799 \cdot 811 \cdot 817 + 320.
\end{align*}
\]

She notes that two of them have exactly $8$ positive divisors each. Find the common prime divisor of those two integers.

Let's think step by step and output the final answer within \boxed{}.
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