1. A square with a side length of $12 \mathrm{~cm}$ is given. Find the dimensions of all different rectangles whose side lengths are different natural numbers, and whose area is equal to the area of the given square. Which of the obtained rectangles has the largest, and which has the smallest perimeter?
Solution. The rectangle has the same area as the given square, so the area of the rectangle is $144 \mathrm{~cm}^{2}$. We factorize the number 144 into divisors, i.e., $144=2^{4} \cdot 3^{2}$, and form all possible pairs of different natural numbers whose product is 144. This gives us 6 rectangles with sides: 2 and 72, 4 and 36, 6 and 24, 8 and 18, 3 and 48, 9 and 16, expressed in $\mathrm{cm}$.

The perimeter of each of them is

$$
\begin{aligned}
& 2(2+72)=148 \mathrm{~cm}, 2(4+36)=80 \mathrm{~cm}, 2(6+24)=60 \mathrm{~cm} \\
& 2(8+18)=52 \mathrm{~cm}, 2(3+48)=102 \mathrm{~cm}, 2(9+16)=50 \mathrm{~cm}
\end{aligned}
$$

The rectangle with the largest perimeter has sides $2 \mathrm{~cm}$ and $72 \mathrm{~cm}$, and the one with the smallest has sides $9 \mathrm{~cm}$ and $16 \mathrm{~cm}$.