GAUSS7 2024 Problem 23

A rectangular prism has integer edge lengths and has a volume of V. The six faces of the prism are painted and then the prism is cut into 1 by 1 by 1 cubes. Of these cubes, 50 cubes have no paint on them. What is the mean (average) of all possible values of V?

Answer Choices
A. 224
B. 310
C. 396
D. 288
E. 348