jmc

Let $l$ be the length and $w$ be the width of the rectangular prism. Then, the surface area of the prism is given by $$2lw+2l\cdot 3+2w\cdot 3=2lw+6l+6w=52.$$Dividing through by $2$, we obtain $lw+3l+3w=26$, and using Simon's Favorite Factoring Trick, it follows that $$lw+3l+3w+9=(l+3)(w+3)=35.$$The pairs of (positive) factors of $35$ are given by $\{1,35\},\{5,7\}$. Only the latter will work, which will give $\{l,w\}=\{2,4\}$. It follows that the volume of the prism is given by $2\times 4\times 3=\boxed{24}$.