8.3. Workers were laying a floor of size $n \times n$ using two types of tiles: $2 \times 2$ and $3 \times 1$. It turned out that they managed to completely cover the floor such that the same number of tiles of each type was used. For which $n$ could this have been possible? (Cutting tiles or overlapping them is not allowed.)
Answer. For $n$ divisible by 7.

Solution. Let the workers use $x$ tiles of each type. Then the area covered by the tiles is $4x + 3x = 7x = n^2$. Therefore, $n$ must be divisible by 7.

If $n$ is divisible by 7, then the floor can be laid. It is sufficient to note that a $7 \times 7$ square can be laid using 7 tiles of each type (see Fig. 5). And a $7k \times 7k$ square can be divided into $7 \times 7$ squares.

Comment. It is shown that $n$ is divisible by $7$ - 3 points.

![](https://cdn.mathpix.com/cropped/2024_05_06_57489c3b6cd1549a2206g-2.jpg?height=268&width=263&top_left_y=370&top_left_x=1028)

Fig. 5

It is shown that for $n=7k$ the floor can be laid - 4 points.