19. One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?
A 27
B 64
C 125
D 216
E 512
19. D The first six positive cubes are $1,8,27,64,125,216$. Clearly, 64 cannot be the sum of three positive cubes as the sum of all the positive cubes smaller than 64 is $1+8+27=36$. Similarly, 125 cannot be the sum of three positive cubes as the largest sum of any three positive cubes smaller than 125 is $8+27+64=99$. However, we note that $27+64+125=216$, so 216 is the smallest cube which is the sum of three positive cubes.