Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$.  One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home.  How old is Joey?

$\textbf{(A)}\ 3 \qquad\textbf{(B)}\ 7  \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 13$
Because the $5$-year-old stayed home, we know that the $11$-year-old did not go to the movies, as the $5$-year-old did not and $11+5=16$.  Also, the $11$-year-old could not have gone to play baseball, as he is older than $10$.  Thus, the $11$-year-old must have stayed home, so Joey is $\boxed{\textbf{(D) }11}$