Rectangles with integer dimensions

Problem:

Find all rectangles with integer dimensions such that its perimeter is equals to its area.

Solution:

Let the length and width of the rectangle be $x$ and $y$ respectively. We have

$2(x + y) = xy$

Of course, subject to the constraints that $x \ge 0$, $y \ge 0$, $x, y \in \mathbf{Z}$.

The key idea is to factorize this as follow:

$xy - 2x - 2y + 4 = 4$
$(x - 2)(y - 2) = 4$

Once we have that, we know there are only a handful of integer factorization of 4, and we can now check:

$(x - 2) = 1, (y - 2) = 4$ gives $x = 3$, $y = 6$
$(x - 2) = 2, (y - 2) = 2$ gives $x = 4$, $y = 4$

So there we go, there are only two rectangles that satisfy the requirements.