There are exactly $N$ distinct rational numbers $k$ such that $|k|<200$ and \[5x^2+kx+12=0\] has at least one integer solution for $x$.  What is $N$?
$\textbf{(A) }6\qquad \textbf{(B) }12\qquad \textbf{(C) }24\qquad \textbf{(D) }48\qquad \textbf{(E) }78\qquad$
Factor the quadratic into
\[\left(5x + \frac{12}{n}\right)\left(x + n\right) = 0\]
where $-n$ is our integer solution. Then, 
\[k = \frac{12}{n} + 5n,\]
which takes rational values between $-200$ and $200$ when $|n| \leq 39$, excluding $n = 0$. This leads to an answer of $2 \cdot 39 = \boxed{\textbf{(E) } 78}$.