A rectangular room is 6 meters 24 centimeters long and 4 meters 32 centimeters wide. The floor must be covered completely using square tiles of equal size (no cutting and no gaps). What is the least number of such square tiles required to cover the entire floor?

Introduction / Context:
This is a standard tiling problem that uses HCF. To cover a rectangular floor with square tiles of equal size without cutting, the side length of each tile must divide both the length and width of the room exactly. To minimize the number of tiles, we choose the largest such tile, which corresponds to the HCF of the two dimensions.

Given Data / Assumptions:

• Length of room = 6 m 24 cm = 624 cm
• Width of room = 4 m 32 cm = 432 cm
• Tiles are square and all identical
• No cutting or gaps are allowed

Concept / Approach:
Let the side of each square tile be s cm. Then s must be a common divisor of 624 and 432. To minimize the number of tiles, s should be equal to the HCF of 624 and 432. Once s is known, the number of tiles is:
Number of tiles = (624 / s) * (432 / s)

Step-by-Step Solution:
Step 1: Compute gcd(624, 432).624 mod 432 = 192.432 mod 192 = 48.192 mod 48 = 0, so gcd(624, 432) = 48.Step 2: Therefore, the side of each square tile is s = 48 cm.Step 3: Number of tiles along the length = 624 / 48 = 13.Step 4: Number of tiles along the width = 432 / 48 = 9.Step 5: Total number of tiles = 13 * 9 = 117.

Verification / Alternative check:
If a larger tile size than 48 cm were chosen, it would not divide both 624 and 432 exactly, making tiling without cutting impossible. Any smaller tile size that still divides both dimensions, such as 24 cm, would lead to more tiles (26 by 18 = 468 tiles), which is not minimal. Thus, using tiles of side 48 cm and a total of 117 tiles is optimal.

Why Other Options Are Wrong:
107, 127, 137, 156: None of these counts correspond to a valid square tile size that divides both 624 cm and 432 cm and also yields a minimal number of tiles. They represent either smaller tile sizes or tilings that are not consistent with the HCF-based calculation.

Common Pitfalls:
Not converting meters and centimeters into a single unit before finding the HCF.Using LCM instead of HCF for the tile side, which would give a very large and impractical result.Forgetting to multiply the number of tiles along the length and width to obtain the total.

Final Answer:
117