A rectangular block, with dimensions $4 \mathrm{~cm}, 5 \mathrm{~cm}$ and $6 \mathrm{~cm}$, is made up of cubes each with side length $1 \mathrm{~cm}$. If $1 \mathrm{~cm}^{3}$ cubes are removed from this larger rectangular block, what is the minimum number of these cubes that must be removed so that the resulting solid is itself a cube?
(A) 40
(B) 93
(C) 46
(D) 64
(E) 56
The original rectangular block measures $5 \mathrm{~cm}$ by $6 \mathrm{~cm}$ by $4 \mathrm{~cm}$, and so has volume $5 \times 6 \times 4=120 \mathrm{~cm}^{3}$. This tells us that it is made up of 120 of the small cubes. The largest cube that can be formed by removing cubes is a $4 \mathrm{~cm}$ by $4 \mathrm{~cm}$ by $4 \mathrm{~cm}$ cube, since each side length can be no more longer than the side lengths of the original block. This new cube is made up of $4 \times 4 \times 4=64$ of the $1 \mathrm{~cm}^{3}$ cubes, and so 56 of the original $1 \mathrm{~cm}^{3}$ cubes have been removed.

ANSWER: (E)