A room measures 5.44 m by 3.75 m and is to be paved with identical square tiles (no cutting). Find the least number of tiles required to exactly cover the floor.

Introduction / Context:
When covering a rectangular floor with identical square tiles without cutting, the largest possible tile side that fits both dimensions exactly must be used to minimize the number of tiles. This side length equals the greatest common divisor (GCD) of the two side lengths, expressed in the same unit with integers (usually centimeters). Then, the number of tiles equals the area of the floor divided by the area of one tile.

Given Data / Assumptions:

• Room: 5.44 m by 3.75 m.
• Convert to centimeters: 544 cm by 375 cm.
• Square tiles must tile both sides exactly (no cutting).

Concept / Approach:
Find tile side = GCD(544, 375) (in cm). Because the two lengths are coprime (no common factor > 1), the GCD is 1 cm. Thus, the largest square tile that exactly fits both dimensions has side 1 cm. The least number of tiles then equals floor area in cm^2.

Step-by-Step Solution:
Compute GCD(544, 375): successive remainders → … → 1 ⇒ GCD = 1 cmTile side = 1 cm ⇒ tile area = 1 cm^2Floor area = 544 * 375 = 204,000 cm^2Number of tiles = 204,000 / 1 = 204,000

Verification / Alternative check:
Any larger tile side (e.g., 2 cm) fails to divide at least one dimension (375). Therefore, 1 cm is the largest exact-fitting square side, which gives the minimum count.

Why Other Options Are Wrong:
203,500, 205,000, and 192,000 do not equal the exact area 204,000 cm^2 obtainable only with the maximal exact tile side 1 cm.

Common Pitfalls:
Confusing “least number of tiles” with “smallest tile side”; here, using the GCD produces the largest allowable tile side that still fits, which minimizes the count.

Final Answer:
204,000