If length and breadth of a room is 15 m 17 cm and 9 m 2 cm respectively. What…

To tile a rectangular floor completely with the fewest identical square tiles, the tile's side must equal the Highest Common Factor (HCF) of the room's length and breadth, measured in the same unit. The HCF is the largest length that divides both dimensions exactly, so any bigger tile would leave a gap along at least one side.

1. Convert both dimensions to the same unit (centimetres): Length = 15 m 17 cm = (15 x 100) + 17 = 1517 cm; Breadth = 9 m 2 cm = (9 x 100) + 2 = 902 cm.

2. Find the HCF of 1517 and 902 using the Euclidean algorithm: 1517 = 1 x 902 + 615; 902 = 1 x 615 + 287; 615 = 2 x 287 + 41; 287 = 7 x 41 + 0. So HCF(1517, 902) = 41 cm - this is the side of the largest square tile that fits both dimensions exactly.

3. Find how many tiles fit along each side: Length / tile side = 1517 / 41 = 37 tiles; Breadth / tile side = 902 / 41 = 22 tiles.

4. Total minimum number of tiles = tiles along length x tiles along breadth = 37 x 22 = 814.

Cross-check: 41 x 37 = 1517 and 41 x 22 = 902, confirming 41 divides both dimensions exactly with no remainder. The prime factorisations confirm 41 is indeed the highest common factor: 1517 = 37 x 41 and 902 = 2 x 11 x 41, whose only common factor greater than 1 is 41.

Hence, the minimum number of square tiles required is 814.