## Task 4

The lengths of all sides of the rectangle, expressed in centimeters, are natural numbers. If we increase the width and height of this rectangle by $10 \mathrm{~cm}$, its area will double. Determine the dimensions of all such rectangles.
## Second Solution.

$$
\begin{gathered}
(a+10) \cdot(b+10)=2 \cdot a \cdot b \\
a b+10 a+10 b+100=2 a b \\
a b-10 a=10 b+100 \\
a(b-10)=10 b+100 \\
a=\frac{10 b+100}{b-10}=\frac{10 b-100+200}{b-10}=10+\frac{200}{b-10}
\end{gathered}
$$

Thus, $b-10$ must be a divisor of the number $200=2^{3} \cdot 5^{2}$. For each such $b$, we calculate the corresponding $a$ and obtain 3 solutions:

| $b-10$ | 1 | 2 | 4 | 5 | 8 | 10 | 20 | 25 | 40 | 50 | 100 | 200 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $b$ | 11 | 12 | 14 | 15 | 18 | 20 | 30 | 35 | 50 | 60 | 110 | 210 |
| $a$ | 210 | 110 | 60 | 50 | 35 | 30 | 20 | 18 | 15 | 14 | 12 | 11 |

The possible lengths of the sides of the original rectangle are:
I. $\quad 11 \mathrm{~cm}$ and $210 \mathrm{~cm}$

II. $\quad 12 \mathrm{~cm}$ and $110 \mathrm{~cm}$

III. $\quad 14 \mathrm{~cm}$ and $60 \mathrm{~cm}$

IV. $\quad 15 \mathrm{~cm}$ and $50 \mathrm{~cm}$
V. $\quad 18 \mathrm{~cm}$ and $35 \mathrm{~cm}$

VI. $\quad 20 \mathrm{~cm}$ and $30 \mathrm{~cm}$