The problem asks for the number of tiles, n, needed to pave the smallest possible square floor using tiles of size 8 units by 6 units without any breakage.

Finding the Smallest Square Floor Dimension

To pave a square floor completely with $8 \times 6$ tiles without breaking them, the side length of the square floor must be a multiple of both the tile's length (8) and width (6).

We need to find the smallest common multiple (LCM) of 8 and 6.

• Prime factorization of 8: $2^3$
• Prime factorization of 6: $2 \times 3$
• LCM(8, 6) = $2^3 \times 3 = 8 \times 3 = 24$

Therefore, the side length of the smallest square floor that can be paved is 24 units.

Calculating Areas

Calculate the area of the smallest square floor:

Area of square floor = side $\times$ side = $24 \times 24 = 576$ square units.

Calculate the area of a single tile:

Area of one tile = length $\times$ width = $8 \times 6 = 48$ square units.

Calculating the Number of Tiles (n)

The total number of tiles needed, n, is the area of the square floor divided by the area of one tile.

$ n = \frac{\text{Area of square floor}}{\text{Area of one tile}} $

$ n = \frac{576}{48} $

$ n = 12 $

Thus, 12 tiles are needed.