2021 Gauss 8 Problem 25

Square PQRS has sides of length 8. It is split into four rectangular regions by two line segments, one parallel to PQ and another parallel to QR. There are N ways in which these lines can be drawn so that the area of each of the four rectangular regions is a positive integer. What is the remainder when N^2 is divided by 100?

\textbf{(A)}\ 9\quad \textbf{(B)}\ 61\quad \textbf{(C)}\ 1\quad \textbf{(D)}\ 41\quad \textbf{(E)}\ 36

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.