A rectangular floor in my office has its area equal to `56 m^2`. The minimum number of tiles required, if all the tiles are in square shape is:

Options

A. 15
B. 9
C. 14
D. Can't be determined

Answer

To find the minimum number of square tiles required to cover a rectangular floor with an area of 56 m², we can follow these steps:

### Step 1: Understand the Area of the Rectangle
The area of a rectangle is given by the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
In this case, the area is provided as 56 m².

### Step 2: Define the Area of a Square Tile
Let \( a \) be the side length of one square tile. The area of one square tile is:
\[ \text{Area of one tile} = a^2 \]

### Step 3: Set Up the Equation
If \( n \) is the number of square tiles required to cover the rectangular floor, then the total area covered by the tiles can be expressed as:
\[ n \times a^2 = 56 \]

### Step 4: Rearranging the Equation
From the equation above, we can express \( n \) in terms of \( a \):
\[ n = \frac{56}{a^2} \]

### Step 5: Minimize the Number of Tiles
To minimize \( n \), we need to maximize \( a^2 \). The largest square tile that can fit into the area of 56 m² will be when \( a^2 \) is a factor of 56. The factors of 56 are:
1. 1
2. 2
3. 4
4. 7
5. 8
6. 14
7. 28
8. 56

### Step 6: Calculate Minimum Tiles for Each Factor
Now we will calculate \( n \) for each factor:
- For \( a^2 = 1 \): \( n = \frac{56}{1} = 56 \)
- For \( a^2 = 2 \): \( n = \frac{56}{2} = 28 \)
- For \( a^2 = 4 \): \( n = \frac{56}{4} = 14 \)
- For \( a^2 = 7 \): \( n = \frac{56}{7} = 8 \)
- For \( a^2 = 8 \): \( n = \frac{56}{8} = 7 \)
- For \( a^2 = 14 \): \( n = \frac{56}{14} = 4 \)
- For \( a^2 = 28 \): \( n = \frac{56}{28} = 2 \)
- For \( a^2 = 56 \): \( n = \frac{56}{56} = 1 \)

### Step 7: Conclusion
The minimum number of square tiles required is when \( a^2 = 56 \), which means only 1 tile is needed.

Thus, the minimum number of tiles required is:
\[ \boxed{1} \]