Suppose m and n are positive odd integers. Which of the following must also be an odd integer?
$\textbf{(A)}\ m+3n\qquad\textbf{(B)}\ 3m-n\qquad\textbf{(C)}\ 3m^2 + 3n^2\qquad\textbf{(D)}\ (nm + 3)^2\qquad\textbf{(E)}\ 3mn$
Assume [WLOG](https://artofproblemsolving.com/wiki/index.php/WLOG) that $m$ and $n$ are both $1$. Plugging into each of the choices, we get $4, 2, 6, 16,$ and $3$. The only odd integer is $\boxed{\textbf{(E)}\ 3mn}$.