For how many positive integers $m$ does there exist at least one positive integer n such that $m \cdot n \le m + n$?
$\textbf{(A) } 4\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 12\qquad \textbf{(E) } \text{infinitely many}$
For any $m$ we can pick $n=1$, we get $m \cdot 1 \le m + 1$,
therefore the answer is $\boxed{\textbf{(E) } \text{infinitely many}}$.