2007 iTest Problems/Problem 43

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Problem

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 \$ (Error compiling LaTeX. Unknown error_msg) $9$-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.

Solution