The perimeter is equal to the area The measurements on the sides of a rectangle are distinct integers. The perimeter and area of ​​the rectangle are expressed by the same number. Determine this number. 
Answer: 18 
It could be $4*4$ = $4+4+4+4$ but the answer is 18. 
Wait... Now that I noticed, the sides are different numbers. But I can't find a way to solve.

Let $x$ and $y$ be the sides of the rectangle.  Then the given condition implies that $xy = 2x + 2y$.  Solving for $y$ in terms of $x$, we get:
$$y = \frac{2x}{x-2}.$$
It's not immediately obvious from this equation for what integer values of $x$ this gives an integer value of $y$.  However, by doing a polynomial division, we can rewrite this as:
$$y = 2 + \frac{4}{x-2}.$$
Now, it is clear that $y$ is an integer if and only if $x-2$ is a factor of 4.  The integers factors of 4 are $\pm 4, \pm 2, \pm 1$; however, since $x$ must be positive, we can eliminate the cases $x-2 = -4$ and $x-2 = -2$.  For the other factors, plugging into the equation we get solutions $(x,y) = (1,-2), (3, 6), (4, 4), \mathrm{or}~(6,3)$.  However, $y$ must also be positive, eliminating the $(1,-2)$ solution; and we are given $x \ne y$, eliminating the $(4, 4)$ solution.  Therefore, $(x,y) = (3,6)~\mathrm{or}~(6,3)$, and in both cases, $xy = 2x + 2y = 18$.
