1996 AHSME Problem 27

Consider two solid spherical balls, one centered at (0, 0,\frac{21}{2}) with radius 6, and the other centered at (0, 0, 1) with radius \frac{9}{2}. How many points with only integer coordinates (lattice points) are there in the intersection of the balls?

\text{(A)}\ 7\qquad\text{(B)}\ 9\qquad\text{(C)}\ 11\qquad\text{(D)}\ 13\qquad\text{(E)}\ 15

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.