Problem:

We define a chessboard polygon to be a polygon whose edges are situated along lines of the form $x = a$ and $y = b$, where $a$ and $b$ are integers. These lines divide the interior into unit squares, which we call cells.

Let $n$ and $k$ be positive integers. Assume that a square can be partitioned into $n$ congruent chessboard polygons of $k$ cells each. Prove that this square may also be partitioned into $k$ congruent chessboard polygons of $n$ cells each.