2003 Fermat Problem 25

The set \{1,4,n\} has the property that when any two distinct elements are chosen and 2112 is added to their product, the result is a perfect square. If n is a positive integer, the number of possible values for n is

\textbf{(A)}\ 8\quad \textbf{(B)}\ 7\quad \textbf{(C)}\ 6\quad \textbf{(D)}\ 5\quad \textbf{(E)}\ 4

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 Fermat contests here.