To find the minimum number of square tiles needed to cover the entire floor, we follow these steps:

Step 1: Convert the Dimensions to the Same Unit

The given dimensions of the floor are:

• Length = 8 m 96 cm = 896 cm
• Breadth = 6 m 72 cm = 672 cm

Step 2: Find the Largest Possible Square Tile Size

The largest square tile that can be used should have a side length equal to the Greatest Common Divisor (GCD) of 896 cm and 672 cm.

Find the GCD of 896 and 672:

Prime factorizations:

• $896 = 2^7 \times 7$
• $672 = 2^5 \times 3 \times 7$

The GCD is determined by the lowest powers of common factors:
$GCD = 2^5 \times 7 = 32 \times 7 = 224 \text{ cm}$

Step 3: Compute the Number of Tiles

Each square tile has a side length of 224 cm.

• Number of tiles along the length:
  $\frac{896}{224} = 4$
• Number of tiles along the breadth:
  $\frac{672}{224} = 3$

Step 4: Calculate the Total Number of Tiles

$4 \times 3 = 12$

Final Answer:

The minimum number of square tiles required to cover the entire floor is 12.

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Would you like a detailed breakdown of the GCD calculation? Let me know if you have any questions!

Related Questions:

1. How do you find the greatest common divisor (GCD) using prime factorization?
2. What is the formula to find the number of tiles required for any rectangular floor?
3. If the tile size was 100 cm instead, how many tiles would be required?
4. How does the least common multiple (LCM) relate to tiling problems?
5. What happens if the tile size is not a divisor of both dimensions?

Tip:

When solving tiling problems, always check the GCD of the dimensions to find the largest possible square tile, minimizing the number of tiles needed! 🚀