What is the least number of square tiles required to pave completely the floor of a room that is 15 m 17 cm long and 9 m 2 cm broad?

Introduction / Context:
This problem explores the idea of tiling a rectangular floor with identical square tiles so that there is no cutting or wastage. It is common in aptitude exams under the heading of least common multiple and greatest common divisor, applied in a geometric setting. The goal is to find the largest possible tile size that exactly divides both dimensions, and then compute how many such tiles are required.

Given Data / Assumptions:
• Length of the room = 15 m 17 cm.• Breadth of the room = 9 m 2 cm.• Tiles are square and all have the same side length.• Tiles must be placed without overlapping and without leaving gaps.

Concept / Approach:
First, convert both dimensions into centimetres to work with whole numbers. Then, find the greatest common divisor of the two lengths in centimetres. This greatest common divisor is the side of the largest possible square tile that can exactly fit along both length and breadth. The number of tiles will then be equal to the floor area divided by the area of one tile. This approach ensures the least number of tiles, since a larger tile size means fewer tiles.

Step-by-Step Solution:
Step 1: Convert dimensions into centimetres.Length = 15 m 17 cm = 1500 cm + 17 cm = 1517 cm.Breadth = 9 m 2 cm = 900 cm + 2 cm = 902 cm.Step 2: Find the greatest common divisor of 1517 and 902.Using the Euclidean algorithm, gcd(1517, 902) = 41 cm.Step 3: Use 41 cm as the side of the largest possible square tile.Step 4: Compute the number of tiles along the length.1517 / 41 = 37 tiles.Step 5: Compute the number of tiles along the breadth.902 / 41 = 22 tiles.Step 6: Compute the total number of tiles.Total tiles = 37 * 22 = 814.

Verification / Alternative check:
We can confirm that 41 divides both dimensions exactly since 41 * 37 = 1517 and 41 * 22 = 902. No larger square side can divide both 1517 and 902 because 41 is the greatest common divisor. If we check a larger number like 43 or 50, it does not divide at least one of the dimensions exactly. Hence, 41 is the correct tile side and 814 tiles is the minimal count.

Why Other Options Are Wrong:
Option B: 820 would correspond to a different non maximal tile size or to rounding errors.Option C: 840 is too large and assumes smaller tiles, leading to more tiles than necessary.Option D: 844 also indicates a smaller tile side and does not reflect the greatest common divisor approach.

Common Pitfalls:
Students sometimes convert only one dimension into centimetres or make mistakes in converting metres and centimetres. Others look for the least common multiple instead of the greatest common divisor, which is incorrect here because we want the largest tile that fits exactly. Using a systematic method to find the greatest common divisor and carefully converting units helps avoid these typical mistakes.

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