AMC 8 1991 Problem 24

A cube of edge $3$ cm is cut into $N$ smaller cubes, not all the same size. If the edge of each of the smaller cubes is a whole number of centimeters, then $N=$ $\text{(A)}\ 4 \text{(B)}\ 8 \text{(C)}\ 12 \text{(D)}\ 16 \text{(E)}\ 20$