What is the least number of square tiles required to pave (cover completely) the floor of a room that is 15.17 m long and 9.02 m broad?\nAssume tiles are identical squares and there should be no cutting of tiles (use the largest possible square tile).

Introduction:
This question tests the “least number of square tiles” concept, which is fundamentally a greatest common divisor (GCD) problem in disguise. To avoid cutting tiles, the side length of the square tile must exactly divide both the room’s length and breadth. To minimize the number of tiles, we must choose the largest possible tile side that still divides both dimensions. That largest possible tile side is the GCD of the two dimensions (expressed in the same unit). After finding the tile side, the number of tiles is (length/tile_side) * (breadth/tile_side). This is a multi-step aptitude question involving unit conversion and Euclidean algorithm logic.

Given Data / Assumptions:

• Room length = 15.17 m
• Room breadth = 9.02 m
• Use identical square tiles, no cutting allowed
• Convert to cm to avoid decimals: 15.17 m = 1517 cm, 9.02 m = 902 cm
• Largest tile side = GCD(1517, 902)

Concept / Approach:
Find GCD using Euclidean algorithm:\nGCD(a, b) = GCD(b, a mod b) until remainder becomes 0. Then compute number of tiles along each side and multiply to get total tiles. This ensures the tiles fit perfectly and the count is minimal.

Step-by-Step Solution:
Convert: 15.17 m = 1517 cm, 9.02 m = 902 cmCompute GCD(1517, 902)1517 mod 902 = 615902 mod 615 = 287615 mod 287 = 41287 mod 41 = 0, so GCD = 41 cmTiles along length = 1517 / 41 = 37Tiles along breadth = 902 / 41 = 22Total tiles = 37 * 22 = 814

Verification / Alternative Check:
If tile side is 41 cm, then 37 tiles make exactly 37*41 = 1517 cm (15.17 m) and 22 tiles make exactly 22*41 = 902 cm (9.02 m). So the flooring is covered with no cutting and no gaps. Any larger tile side would fail to divide at least one dimension, forcing cutting. Hence 814 is the least possible number of tiles.

Why Other Options Are Wrong:
714, 713, 744, 614: these would correspond to smaller tile sizes or incorrect division steps, meaning either tiles do not fit exactly or the chosen tile is not the largest possible, increasing the count unnecessarily.

Common Pitfalls:
Not converting metres to centimetres and making rounding errors with decimals.Using LCM instead of GCD.Choosing a tile side that divides one dimension but not the other.Mistake in Euclidean algorithm steps or final multiplication 37*22.

Final Answer:
814