## 115. Math Puzzle $12 / 74$

A long rectangular club room is to be prepared.

It was found that the perimeter of the floor area, measured in meters, is numerically equal to its area.

How long are the sides of the room if they are a whole number of meters?
The club room has a length of $x \text{ m}$ and a width of $y \text{ m}$ ($x$ and $y$ are integers). Then it follows that $2(x+y)=xy$

$$
y=\frac{2x}{x-2}=2+\frac{4}{x-2}
$$

Since $x$ and $y$ should be positive and integers, 4 must be divisible by $x-2$ without a remainder, and $x$ must be greater than 2. The possible values are

$$
x=3, y=6 ; \quad x=4, y=4 ; \quad x=6, y=3
$$

The first and third possibilities are the same. Since the room should be elongated, the only suitable dimensions are $x=6 \text{ m}, y=3 \text{ m}$.