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I'm not even sure how to approach this question. At first, I thought that because the flux of the field $F$ through the smaller sphere was $20$, then the flux through the larger sphere should also be $20$ (Does the flux of $F$ represent the rate of flow of gas?) but that doesn't necessarily have to be true.

I'm not sure how to use the divergence either. I think the divergence theorem states that the flux through the larger sphere is equal to the volume integral of the divergence throughout the larger sphere, but I only have the divergence on the surface of the smaller sphere. Am I supposed to calculate the divergence at all points from that information?

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I think I figured it out now.

They give that the divergence everywhere on $E$ is equal to 3, and that the flux through the smaller sphere is $20$. From the divergence theorem, we know that(roughly)

$$\text{Flux}_{\text{bigsphere}} = \int_{\text{bigsphere}} (\nabla \cdot F) dV$$ $$ = \int_{\text{littlesphere}}(\nabla \cdot F) dV + \int_{\text{E}}(\nabla \cdot F) dV$$ $$ = \text{Flux}_{\text{littlesphere}} + \int_{\text{E}}(3) dV$$

So the flux through the big sphere is equal to the flux through the little sphere plus the integral of the divergence of the volume that wasn't accounted for ($E$).

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