A room is 5 m 44 cm long and 3 m 74 cm broad. It is to be paved with identical square tiles such that no tile is cut. Find the least number of square tiles required to cover the floor completely.

Introduction / Context:
This question checks the use of HCF (GCD) to find the largest possible square tile that can exactly cover a rectangular floor without cutting. Once the tile side is found, the least number of tiles is obtained by counting how many tiles fit along the length and breadth and multiplying those counts.

Given Data / Assumptions:

• Length = 5 m 44 cm = 544 cm
• Breadth = 3 m 74 cm = 374 cm
• Tile side = HCF(544, 374) cm
• Least tiles = (544/t) * (374/t)

Concept / Approach:
Convert dimensions to centimetres. Find the HCF (GCD) to get the largest tile side. Then compute number of tiles along each direction and multiply to get total tiles.

Step-by-Step Solution:
Convert length: 5*100 + 44 = 544 cm Convert breadth: 3*100 + 74 = 374 cm HCF(544, 374) = 34 cm Tiles along length = 544/34 = 16 Tiles along breadth = 374/34 = 11 Total tiles = 16*11 = 176

Verification / Alternative check:
A 34 cm tile divides both 544 and 374 exactly, so no cutting is needed. Any larger tile would fail to divide at least one dimension, increasing cutting or making paving impossible under the 'no cut' rule.

Why Other Options Are Wrong:
136 and 146: too few tiles; would imply a larger tile side that does not divide both dimensions exactly. 166: incorrect multiplication or incorrect HCF. 186: would imply different tile counts that do not match 16 by 11 arrangement.

Common Pitfalls:
Not converting metres and centimetres to the same unit. Using LCM instead of HCF for tile size. Dividing only one dimension by tile size and forgetting the other.

Final Answer:
Least number of tiles required = 176