The only possibilities for the numbers are $11,12,13,14,15,16$ and $10,11,12,13,14,15$.  
In the second case, the common sum would be $(10+11+12+13+14+15)/3=25$, so $11$ must be paired with $14$, which
it isn't.  
Thus, the only possibility is the first case and the sum of the six numbers is $81\rightarrow \boxed{\text{E}}$.
The numbers on the faces of this cube are consecutive whole numbers.  The sum of the two numbers on each of the three pairs of opposite faces are equal.  The sum of the six numbers on this cube is

$\text{(A)}\ 75 \qquad \text{(B)}\ 76 \qquad \text{(C)}\ 78 \qquad \text{(D)}\ 80 \qquad \text{(E)}\ 81$