Integrating $\iint \hat{n} dS $ over a closed surface?

Integrating $\iint \hat{n} dS $ over a closed surface?

One of the exercises in the book Div, Grad, Curl, and All That is to show
that $$ \iint_S \hat{n} \hspace{1mm} dS = 0$$ for every closed surface
$S$, using the divergence theorem.
I know the theorem, namely that $$\iint_S {F \cdot \hat{n}} \hspace{1mm}
dS = \iiint_V \nabla \cdot F \hspace{1mm} dV,$$
but I'm not sure how to proceed in this case. What confuses me is the fact
that I'm not integrating a scalar function anymore but a vector
function... And I have no idea what should I do with the right side of the
equation in this case...
Any hints?