A classroom is 6 m 24 cm long and 4 m 32 cm wide. Find the least number of identical square tiles required to exactly cover the entire floor of the classroom.

Introduction / Context:
This question checks your understanding of how to use the highest common factor (HCF) to find the largest possible size of a square tile that can exactly cover a rectangular floor without leaving any gaps or needing to cut tiles. It is a standard application of HCF in mensuration and tiling problems.

Given Data / Assumptions:

• Length of classroom = 6 m 24 cm
• Width of classroom = 4 m 32 cm
• Tiles are square and all of the same size
• We want the least number of tiles, which means the largest possible tile side that fits both dimensions exactly

Concept / Approach:
To minimise the number of tiles, we must maximise the side length of each square tile. That side must be a common divisor of both the length and the width. Therefore, we convert all dimensions into the same unit, find the HCF of the two side lengths, use that as the tile side, and then compute how many tiles are needed in each direction and overall.

Step-by-Step Solution:
Convert 6 m 24 cm to centimetres: 6 m = 600 cm, so length = 600 + 24 = 624 cm Convert 4 m 32 cm to centimetres: 4 m = 400 cm, so width = 400 + 32 = 432 cm Find HCF of 624 and 432 HCF(624, 432) = 48 cm So the largest square tile side = 48 cm Number of tiles along length = 624 / 48 = 13 Number of tiles along width = 432 / 48 = 9 Total number of tiles = 13 * 9 = 117

Verification / Alternative check:
If we chose any tile size larger than 48 cm, it would not divide both 624 and 432 exactly. For example, 50 cm does not divide 432. Any smaller common divisor, such as 24, would lead to more tiles, not fewer. Thus 48 cm gives the least number of tiles and 117 is confirmed as correct.

Why Other Options Are Wrong:
115: Does not correspond to any integer tiling with a common divisor of both sides. 116: This is close to the correct answer but arises from incorrect multiplication or counting. 114: Comes from miscalculating one of the dimensions or tile counts. 130: Significantly higher than the minimum possible and would imply a smaller tile size than 48 cm.

Common Pitfalls:
Forgetting to convert metres to centimetres before taking the HCF. Using the least common multiple (LCM) instead of HCF. Choosing a tile size that divides only one dimension, not both. Making arithmetic mistakes when dividing the dimensions by the tile size.

Final Answer:
The least number of identical square tiles required is 117