Rubik's Cube!

Find the smallest positive integer $k$ , such that for all positive integers $n$ , a $kn \times kn \times kn$ Rubik's cube can be used as a Rubik's cube of any size that is smaller than itself.

• An example of such a cube is a $4 \times 4 \times 4$ cube, because we can use it as a $3 \times 3 \times 3$ cube, a $2 \times 2 \times 2$ cube and a $1 \times 1 \times 1$ cube as shown. The 4 by 4 cube uses all 4 center pieces as one to function like a single center piece on a 3 by 3 cube.