2015 Gauss 8 Problem 24

Two joggers each run at their own constant speed and in opposite directions from one another around an oval track. They meet every 36 seconds. The first jogger completes one lap of the track in a time that, when measured in seconds, is a number (not necessarily an integer) between 80 and 100. The second jogger completes one lap of the track in a time, t seconds, where t is a positive integer. The product of the smallest and largest possible integer values of t is

\textbf{(A)}\ 3705\quad \textbf{(B)}\ 3762\quad \textbf{(C)}\ 2816\quad \textbf{(D)}\ 3640\quad \textbf{(E)}\ 3696

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Gauss contests here.