Answer
$\dfrac{256\pi}{3}$
Work Step by Step
In order to verify the Divergence Theorem which is true for for the vector field over the region $E$, we will have to add all the surface integrals and should be make sure that all are equal to such as: $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, we have $S$ shows a closed surface. The region $E$ is inside that surface.
We have $div F=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z}=1$
The volume of the region $E=\dfrac{4\pi(4)^3}{3}=\dfrac{256\pi}{3}$
Hence, we have $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV =\dfrac{256\pi}{3}$
