Let the total sum be $ S $. The number on each face is added to $ S $ four separate times, since each face borders on $ 4 $ vertices. There are $ 8 $ vertices, and each is the sum of $ 3 $ face numbers since each vertex borders $ 3 $ faces. Thus $ S $ is the sum of $ 8\cdot 3=24 $ face numbers. Since each face is added $ 4 $ times, and there are $ 6 $ faces, we know none of the faces are repeated or left out and each is added exactly $ 4 $ times, so $ S=4(\text{sum of numbers on faces}) $. Thus no matter what the sum of the numbers on the faces is, the total sum $ S $ is always divisible by $ \boxed{4} $.