Well...if the answer isn't 3 or 5, is must be 4, right? Now how do we arrive at this solution...
From mrdexter's work, we know that the only possible combinations are:
1/4/8, 1/5/7, 2/3/8, 2/4/7, 2/5/6
How do we rule out 1/5/7, 2/3/8, and 2/5/6?
Let's go through this again...
2/3/8 ...