A rectangular courtyard 3.78 m long and 5.25 m wide is to be paved completely with square tiles of equal size. What is the largest possible side (in cm) of each square tile?

Introduction / Context:
This question is a practical application of the highest common factor (H.C.F.) concept in the context of tiling a rectangular floor with square tiles. It is a standard type of aptitude problem that tests your ability to convert measurements to consistent units and then use H.C.F. to find the largest possible tile size that divides both length and breadth exactly.

Given Data / Assumptions:

• Length of the courtyard = 3.78 meters.
• Breadth (width) of the courtyard = 5.25 meters.
• The courtyard is to be paved exactly with square tiles of equal side length.
• We are asked to find the largest possible side of a tile in centimetres.

Concept / Approach:
To cover the rectangle exactly with square tiles, the side of each square must be a common divisor of both the length and the breadth. The largest possible tile corresponds to the greatest common divisor of the two dimensions. Since the dimensions are given in meters with decimals, we convert them to centimetres and compute the H.C.F. of the resulting integers. That H.C.F. gives the side of the largest square tile in centimetres.

Step-by-Step Solution:
Step 1: Convert lengths to centimetres: 3.78 m = 378 cm and 5.25 m = 525 cm.Step 2: We need the largest integer that divides both 378 and 525 exactly; this is their H.C.F.Step 3: Find H.C.F. using prime factorization or Euclid's algorithm.Step 4: Factorize: 378 = 2 * 3^3 * 7 and 525 = 3 * 5^2 * 7.Step 5: Common prime factors are 3^1 and 7^1, so H.C.F. = 3 * 7 = 21.Step 6: Therefore, the largest possible side of the square tile is 21 cm.

Verification / Alternative check:
Check divisibility: 378 ÷ 21 = 18, an integer.Also, 525 ÷ 21 = 25, an integer. Since both dimensions are exact multiples of 21 cm and no larger common factor exists, 21 cm is indeed the maximum side length for such square tiles.

Why Other Options Are Wrong:
Option a (14 cms) divides 378 but does not divide 525 exactly with a larger greatest common factor already found. Option c (42 cms) is too large because 378 is not an exact multiple of 42. Option d (None of these) is incorrect because we have found a valid tile size of 21 cms that satisfies all conditions.

Common Pitfalls:
Many students forget to convert metres to centimetres first and directly attempt to compute the H.C.F. of the decimals, which can cause confusion. Others miscalculate the prime factorization or choose a common factor but not the greatest one. Always ensure you work in consistent units and verify that your final tile size divides both dimensions exactly with no remainder.

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