A hall 50m long and 45m broad is to be paved with square tiles. Find the largest tile as well as its number in the given options so that the tiles exactly fit in the hall.

### Concept & Highest Common Factor (HCF) For identical square tiles to fit perfectly across both the length and breadth of a rectangular area without any cutting or breaking, the side length of the tile must be a common factor of both the length and the breadth. To find the *largest* possible tile, you must calculate the Highest Common Factor (HCF) of the room's dimensions. ### Step-by-Step Solution 1. **Given:** Dimensions of the hall are Length ($L$) = 50 m and Breadth ($B$) = 45 m. 2. **Determine the Maximum Tile Side Length:** Find the HCF of 50 and 45. Factors of 50: 1, 2, 5, 10, 25, 50. Factors of 45: 1, 3, 5, 9, 15, 45. The highest common factor is 5. So, the side of the largest square tile is 5 m. 3. **Calculate the Area of One Tile:** Area of tile = $\text{side} \times \text{side} = 5 \times 5 = 25 \text{ sq. m.}$ 4. **Calculate the Total Number of Tiles Required:** Number of tiles = $\frac{\text{Total Area of the Hall}}{\text{Area of One Tile}}$ Number of tiles = $\frac{50 \times 45}{25} = 2 \times 45 = 90 \text{ tiles}$. ### Exam Strategy & Shortcut By inspecting the given options, only one option has a tile area (25 sq. m) whose square root (5 m) perfectly divides both 50 and 45. Options with 36 sq. m (side 6) and 16 sq. m (side 4) cannot divide 45 and 50 perfectly. You can select the correct answer without calculating the total number of tiles. ### Common Pitfall A common mistake is finding the Least Common Multiple (LCM) instead of the HCF, which applies to fundamentally different types of problems (like finding when two runners meet on a track). ### Final Answer Therefore, the correct answer is **25 sq. m and 90 tiles**.