Minimum how many similar tiles of square shape are required to furnish the floor of a room with the length of 462 cm and breadth of 360 cm?

Answer

To determine the minimum number of similar square tiles required to furnish the floor of a room with dimensions 462 cm (length) and 360 cm (breadth), we can follow these steps:

### Step 1: Find the dimensions of the tile
To find the maximum possible dimension of the square tile, we need to calculate the Highest Common Factor (HCF) of the room's length and breadth.

1. **Find the HCF of 462 and 360.**
- The prime factorization of 462 is:
- 462 = 2 × 3 × 7 × 11
- The prime factorization of 360 is:
- 360 = 2^3 × 3^2 × 5
- The common factors are:
- 2 (minimum power is 1) and 3 (minimum power is 1).
- Therefore, HCF = 2 × 3 = 6 cm.

### Step 2: Calculate the area of the floor
Now, we will calculate the area of the floor using the formula:
\[ \text{Area of the floor} = \text{Length} \times \text{Breadth} \]

2. **Calculate the area of the floor:**
- Area = 462 cm × 360 cm = 166320 cm².

### Step 3: Calculate the area of one tile
Next, we will calculate the area of one tile using the side length we found from the HCF.

3. **Calculate the area of one tile:**
- Area of one tile = side × side = 6 cm × 6 cm = 36 cm².

### Step 4: Calculate the number of tiles required
Finally, we can find the number of tiles required by dividing the area of the floor by the area of one tile.

4. **Calculate the number of tiles:**
- Number of tiles = Area of the floor / Area of one tile = 166320 cm² / 36 cm² = 4620 tiles.

### Conclusion
Thus, the minimum number of similar square tiles required to furnish the floor of the room is **4620 tiles**.

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