What is the least number of square tiles required to completely pave the floor of a rectangular room that is 15 metres 17 centimetres long and 9 metres 2 centimetres broad, assuming all tiles are identical and square?

Introduction:
This question involves the concept of tiling a floor with square tiles and using the highest possible tile size so that no cutting is needed. The least number of tiles is achieved when each tile is as large as possible, and its side length is the greatest common divisor (GCD) of the room’s dimensions expressed in the same units. This problem combines knowledge of unit conversion, greatest common divisor, and area-based counting.

Given Data / Assumptions:

• Length of the room = 15 metres 17 centimetres. • Breadth of the room = 9 metres 2 centimetres. • Tiles are square and identical in size. • We want the least number of tiles, so each tile should be as large as possible.

Concept / Approach:
To find the maximum possible side length of the square tile that fits exactly along both dimensions, we convert both dimensions to the same unit (for example, centimetres) and then compute their greatest common divisor (GCD). This GCD is the largest possible tile side length in centimetres. The number of tiles is then the area of the floor divided by the area of one tile, or equivalently the product of (length / tile_side) and (breadth / tile_side).

Step-by-Step Solution:
Step 1: Convert dimensions to centimetres. 1 metre = 100 cm. Length = 15 m 17 cm = 15 * 100 + 17 = 1517 cm. Breadth = 9 m 2 cm = 9 * 100 + 2 = 902 cm. Step 2: Find the greatest common divisor (GCD) of 1517 and 902. Using the GCD concept, gcd(1517, 902) = 41. Step 3: The side of each square tile = 41 cm. Step 4: Find how many tiles fit along each dimension. Number of tiles along length = 1517 / 41 = 37. Number of tiles along breadth = 902 / 41 = 22. Step 5: Compute total number of tiles. Total tiles = 37 * 22 = 814.

Verification / Alternative check:
We can check whether 41 cm really divides both dimensions exactly. 41 * 37 = 1517 and 41 * 22 = 902, so the tile side length fits perfectly into both the length and breadth without leaving any remainder. Thus no cutting of tiles is required, and we are using the largest possible tile size. Therefore, any smaller tile size would increase the total number of tiles, making 814 the minimum possible number.

Why Other Options Are Wrong:
Option 844: This corresponds to a smaller tile size, which would not be the largest possible divisor of both dimensions. Option 840 and 820: These also assume different tile sizes that do not exploit the maximum GCD, leading to more tiles than necessary. Option 780: This is even smaller and cannot be achieved with the maximum tile side, given the exact room dimensions.

Common Pitfalls:
A common mistake is to convert only one dimension to centimetres or to find the least common multiple (LCM) instead of the greatest common divisor. Another pitfall is to guess a tile side that divides one dimension but not the other, which would require cutting and would contradict the requirement for identical, uncut tiles. Always remember to convert both dimensions to the same unit and to use the GCD for such tiling problems.

Final Answer:
The least number of square tiles required is 814.