2015 Gauss 8 Problem 23

The list of integers 4, 4, x, y, 13 has been arranged from least to greatest. How many different possible ordered pairs (x, y) are there so that the average (mean) of these 5 integers is itself an integer?

\textbf{(A)}\ 7\quad \textbf{(B)}\ 8\quad \textbf{(C)}\ 9\quad \textbf{(D)}\ 10\quad \textbf{(E)}\ 11

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.