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 $a$ and $b$ be the sides of the rectangle. We know the perimeter of a rectangle is $ 2(a+b) $, and the area is$(ab)$. 
Since these two expressions have to be equal, let us solve the equation:
$$2(a+b) = ab$$
$$2a = ab - 2b$$
$$2a = b(a-2)$$
$$b = \frac{2a}{a-2}$$
Since $b$ is an integer, $\frac{2a}{a-2}$ must be one as well. This means that $a$ is even, because otherwise when $a$ is odd, $a$ is not divisible by $a-2$, excluding the case $a=3$. This means that $b = \frac{2*3}{3-2} = 6$, which is a valid solution.
We can approach this by doing casework: checking all even numbers from $a=4$ (when $a=2$, $a-2 = 0$).
When $a=4$, $b=\frac{2*4}{4-2}=4$, but this solution does not have distinct integers.
When $a=6$, $b=\frac{2*6}{6-2}=3$. Therefore, this is the same solution mentioned earlier.
Therefore, one solution to the problem is when $a=6,b=3$. Now check if the values of $a$ and $b$ satisfy the original expression.
