A rectangular room has dimensions 6 metres 24 centimetres by 4 metres 32 centimetres. What is the least number of equal square tiles required to completely cover the floor of the room?

Introduction:
This question checks your understanding of using the highest common factor (HCF) to solve geometry and tiling problems. We want to cover a floor with identical square tiles without cutting any tile and without leaving gaps.

Given Data / Assumptions:

• Length of the room = 6 metres 24 centimetres
• Width of the room = 4 metres 32 centimetres
• Tiles are square and all have the same size
• We need the least possible number of tiles

Concept / Approach:
To minimise the number of square tiles, each tile should be as large as possible. The side of the largest possible square that can exactly tile the floor must be the HCF of the two dimensions (expressed in the same unit). Once we know the side of each tile, we divide the area of the floor by the area of one tile to get the number of tiles.

Step-by-Step Solution:
Convert dimensions to centimetres: Length = 6 m 24 cm = 6 × 100 + 24 = 624 cm Width = 4 m 32 cm = 4 × 100 + 32 = 432 cm Find HCF of 624 and 432: 624 = 2^4 × 3 × 13 432 = 2^4 × 3^3 Common prime factors with smallest powers: 2^4 × 3 = 16 × 3 = 48 So side of each largest square tile = 48 cm Number of tiles along length = 624 / 48 = 13 Number of tiles along width = 432 / 48 = 9 Total tiles required = 13 × 9 = 117

Verification / Alternative check:
We can check by multiplication: 117 × (48 × 48) must equal 624 × 432. Since the ratios along length and width are integers (13 and 9), the tiling is exact and no cutting is needed. This confirms the correctness of the HCF based approach.

Why Other Options Are Wrong:
107, 127, and 137 are just random numbers around the correct value and do not come from the factorisation. They would correspond to non integer tile dimensions, which would not fit the room exactly when using square tiles of equal size.

Common Pitfalls:
Many learners forget to convert metres into centimetres before finding the HCF. Another mistake is using the least common multiple (LCM) instead of HCF, which would give incorrect dimensions and far too many tiles. Always remember that maximum tile size corresponds to the HCF of the two side lengths.

Final Answer:
The minimum number of equal square tiles required to cover the room exactly is 117.