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.
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