Remember

Square Divisions? This problem demonstrates the deconstruction of a square into smaller squares with integer-length sides.

Given a cube with edge length 60, can you find a deconstruction of the cube into smaller cubes (none of which are alike) with integer length sides (or prove it can't be done)?

I don't see a solution to this problem.

Regardless of the size of the smallest cube, there remains the problem of lining up in 3 dimensions other sized cubes around this cube without creating gaps. (Even if you stick it in the corner of the 60 x 60 cube.)