Find the sum of all integers $n$ such that $\dfrac{12}{n}$ is also an integer.
The positive factors of $12$ are $1, 2, 3, 4, 6,$ and $12$. The negative factors of $12$ are $-1, -2, -3, -4, -6,$ and $-12$. When we add these twelve numbers together, we can pair each positive factor with its negation: \begin{align*}
[1+(-1)] + [2 + (-2)] + [3 + (-3)]& + [4 + (-4)] \\
{}+ [6 + (-6)] + [12 + (-12)] &= 0 + 0 + 0 + 0 + 0 + 0 \\
&= \boxed{0}.
\end{align*}