Janka and Hanka like to play with animal models. Hanka built a rectangular pen for her little cows using bottle caps from PET bottles, as shown in the picture. Janka used all her bottle caps to build a pen in the shape of an equilateral triangle for her sheep. Then she took it apart and built a square pen, also using all her bottle caps. How many bottle caps could Janka have had? Find at least 2 solutions.

(M. Volfová)

![](https://cdn.mathpix.com/cropped/2024_04_17_0d982c5cbb35b07eed65g-6.jpg?height=265&width=416&top_left_y=747&top_left_x=820)

Hint. Janka could have had, for example, 6 or 8 bottle caps.
That Janka built a fence in the shape of an equilateral triangle means that the number of her closures had to be a multiple of the number 3. Similarly, she could only build a square fence if the number of closures was a multiple of the number 4. The number of closures, therefore, had to be a multiple of both the number 3 and 4, i.e., 12, 24, 36, . (any multiple of 12).