2016 HMMT November General Round Problem 3

Let V be a rectangular prism with integer side lengths. The largest face has area 240 and the smallest face has area 48. A third face has area x, where x is not equal to 48 or 240. What is the sum of all possible values of x?

Answers are checked by value, so any equivalent form is accepted: 1/2, \frac{1}{2} and 0.5 all count as the same answer.

Full credit goes to HMMT for authoring these problems. This problem is from the November 2016 contest; the official solution is available on the HMMT archive. HMMT is not affiliated with or endorsing TopsOJ in any way.