## Task 1 - 310611

Uwe and Jan each draw a rectangle that can be divided into exactly 60 squares, each with a side length of $1 \mathrm{~cm}$. Jan's rectangle has twice the perimeter of Uwe's rectangle.

Determine the side lengths of the rectangles drawn by Uwe and Jan!
By listing all representations of 60 as a product of two natural numbers, one obtains:

There are exactly the following possibilities for a rectangle that can be divided into exactly 60 squares, each with a side length of $1 \mathrm{~cm}$:

| Side lengths in $\mathrm{cm}$ | Perimeter in $\mathrm{cm}$ |
| :---: | :---: |
| 1,60 | 122 |
| 2,30 | 64 |
| 3,20 | 46 |
| 4,14 | 38 |
| 5,12 | 34 |
| 6,10 | 32 |

Of these perimeters, exactly one is twice as large as one of the others, namely $64 \mathrm{~cm}$ is twice as large as $32 \mathrm{~cm}$. Therefore, Jan's rectangle has side lengths of $2 \mathrm{~cm}, 30 \mathrm{~cm}$ and Uwe's rectangle has side lengths of $6 \mathrm{~cm}, 10 \mathrm{~cm}$.

## Solution adopted from $[5]$