A rectangular courtyard is 3.78 meters long and 5.25 meters wide. It is to be paved exactly with square tiles, all of the same size. What is the largest possible side length of each square tile in centimeters?

Introduction / Context:
This is a classic highest common factor (HCF) or greatest common divisor problem disguised as a geometry and tiling question. The idea is to cover a rectangular courtyard fully with square tiles of equal size, without cutting any tile. The largest possible tile side corresponds directly to the greatest common divisor of the courtyard dimensions expressed in the same unit.

Given Data / Assumptions:

• Length of the courtyard = 3.78 meters.
• Width of the courtyard = 5.25 meters.
• Tiles are square and identical.
• We want the largest possible side length of the tile in centimeters.

Concept / Approach:
To find the largest square tile that can exactly pave the rectangle, we:

• Convert all dimensions to the same unit (here, centimeters).
• Find the greatest common divisor of the length and the width in centimeters.
• The HCF will be the largest side length of the square tile that divides both dimensions exactly.

Step-by-Step Solution:
Step 1: Convert 3.78 meters to centimeters: 3.78 * 100 = 378 cm. Step 2: Convert 5.25 meters to centimeters: 5.25 * 100 = 525 cm. Step 3: We need the HCF of 378 and 525. Step 4: Factor 378: 378 = 2 * 3 * 3 * 3 * 7 (or 2 * 3^3 * 7). Step 5: Factor 525: 525 = 3 * 5 * 5 * 7 (or 3 * 5^2 * 7). Step 6: Identify common prime factors: both have one 3 and one 7. Step 7: Multiply the common factors: HCF = 3 * 7 = 21. Step 8: Therefore, the largest possible tile side is 21 centimeters.

Verification / Alternative Check:
Check that 21 cm divides both dimensions exactly: 378 ÷ 21 = 18 tiles along the length, and 525 ÷ 21 = 25 tiles along the width. Both results are whole numbers, so 21 cm tiles will cover the courtyard exactly without cutting. If we try a larger size like 42 cm, 378 ÷ 42 = 9 is an integer but 525 ÷ 42 is not an integer, confirming that 42 cm is too large.

Why Other Options Are Wrong:
14 cm divides 378 and 525, but it is not the greatest common divisor; a larger size, 21 cm, still works.
42 cm does not divide 525 exactly, so the width cannot be tiled perfectly with 42 cm squares.
Thus, the correct largest size is 21 cm, so the option “None of these” is also incorrect.

Common Pitfalls:
One common mistake is failing to convert dimensions to the same units before finding the HCF. Another is guessing tile sizes from the options without verifying exact divisibility. Always compute the highest common factor in such questions to ensure you have the largest possible tile size that fits both dimensions exactly.

Final Answer:
The largest possible side length of each square tile is 21 cms.