jmc

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 $\boxed{4}$.