2014 AMC 8 Problem 13

If n and m are integers and n^2+m^2 is even, which of the following is impossible?

\textbf{(A) } n and m are even \qquad\textbf{(B) } n and m are odd \qquad\textbf{(C) } n+m is even \qquad\textbf{(D) } n+m is odd \qquad \textbf{(E) } none of these are impossible

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.