5.4. A square is cut into four equal squares. Each of them is cut into 7, 8, 9, and 10 equal rectangles as shown in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_8c9b9114972c62456a9cg-05.jpg?height=409&width=412&top_left_y=2234&top_left_x=822)

The lengths of the sides of all rectangles are integers. Find the smallest possible length of the side of the original square.

Answer, option 1. 1008.

Answer, option 2. 3960.

Answer, option 3. 1260.

Answer, option 4. 5040.
Solution Option 1. Evaluation. The side length of each of the four squares is divisible by 6, 7, 8, and 9, which means it is divisible by $\operatorname{LCM}(6,7,8,9)=504$. Therefore, it is not less than 504. Consequently, the side of the original square is not less than 1004.

Example. A square with a side length of 1004 can be cut in the required manner. We cut it into four squares $504 \times 504$, and then cut these into 6 rectangles $504 / 6 \times 504$, into 7 rectangles $504 / 7 \times 504$, into 8 rectangles $504 / 8 \times 504$, and into 9 rectangles $504 / 9 \times 504$.