Four, there is a rectangular prism composed of $m n k$ unit cubes, with its length, width, and height being $m, n$, and $k$ units respectively, and $m \geqslant n \geqslant k$. The three faces of such a rectangular prism containing vertex $A$ are colored, such that exactly half of the unit cubes have none of their faces colored. How many unit cubes in such a rectangular prism have at least one face colored?



Ask the
number of
unit cubes
with at
least one
face
colored in
such a
rectangular
prism.
Four, Hint: The key to this problem lies in listing the relationship between $m, n, k$: $\frac{m n k}{2}=(m-1)(n-1)(k-1)$, and then discussing for $k=3,4,5, \cdots$. The number of them is
$$
\frac{m n k}{2}=60,72,84,90,120 .
$$