Finding Minimum Square Slabs for Paving

To find the minimum number of identical square slabs needed to pave a rectangular floor, we need to determine the largest possible side length for the square slab. This side length must perfectly divide both the length and the breadth of the room.

Calculating Largest Square Slab Size

The largest possible side length for the square slab is the Greatest Common Divisor (GCD) of the room's dimensions.

• Room Length = $4.5 \text{ m}$
• Room Breadth = $1.5 \text{ m}$

We need to find the GCD of $4.5$ and $1.5$.

Notice that $4.5$ is exactly $3$ times $1.5$. Therefore:

$ \text{GCD}(4.5, 1.5) = 1.5 \text{ m} $

So, the largest possible identical square slab that can be used has a side length of $1.5 \text{ m}$.

Calculating Number of Slabs

Now, we calculate how many slabs fit along the length and breadth of the room.

• Number of slabs along the length = $\frac{\text{Room Length}}{\text{Slab Side Length}} = \frac{4.5 \text{ m}}{1.5 \text{ m}} = 3$
• Number of slabs along the breadth = $\frac{\text{Room Breadth}}{\text{Slab Side Length}} = \frac{1.5 \text{ m}}{1.5 \text{ m}} = 1$

The total minimum number of slabs is the product of the number of slabs along the length and breadth.

$ \text{Total Slabs} = (\text{Number along length}) \times (\text{Number along breadth}) $

$ \text{Total Slabs} = 3 \times 1 = 3 $

Therefore, the minimum number of identical square slabs needed is 3.