Difficulty: Hard
Correct Answer: 207570
Explanation:
Introduction / Context:
This question tests the idea of using the greatest possible square tile so that the number of tiles is minimized and no cutting is needed. The side of the largest square tile must exactly divide both the room's length and breadth. Therefore, the tile side is the HCF (GCD) of the two dimensions (after converting to the same unit). Then the number of tiles is (length/tile_side) * (breadth/tile_side).
Given Data / Assumptions:
Concept / Approach:
Convert all lengths to centimetres first. Compute HCF (GCD) to get the largest possible tile side. Then compute how many tiles fit along each dimension, and multiply to get total tiles required.
Step-by-Step Solution:
Convert: 5 m 55 cm = 5*100 + 55 = 555 cm
Convert: 3 m 74 cm = 3*100 + 74 = 374 cm
Compute HCF(555, 374). Since 555 and 374 share no common factor greater than 1, HCF = 1 cm
So the largest tile is 1 cm by 1 cm
Tiles along length = 555/1 = 555
Tiles along breadth = 374/1 = 374
Total tiles = 555*374 = 207570
Verification / Alternative check:
Check divisibility: a 1 cm tile obviously divides any whole centimetre measurement. Because indicating 'least number' forces us to use the largest tile, and the HCF is 1 cm here, there is no larger square tile that will fit both dimensions exactly without cutting.
Why Other Options Are Wrong:
205740, 210000, 198000, 220590: these totals do not equal 555*374 and would correspond to incorrect dimensions or using a tile size that does not exactly divide both 555 and 374.
Common Pitfalls:
Not converting metres and centimetres into the same unit before taking HCF.
Using LCM instead of HCF.
Assuming a 'nice' tile size without checking exact divisibility.
Final Answer:
Least number of tiles required = 207570
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