Compute the H.C.F. (greatest common divisor) of the integers 132, 204, and 228.

Introduction / Context:
We seek the highest common factor (HCF) of three integers: 132, 204, and 228. The HCF is the largest positive integer that divides each of the numbers exactly, leaving no remainder.

Given Data / Assumptions:

• Numbers: 132, 204, 228.
• Standard divisibility and Euclidean algorithm apply.

Concept / Approach:
Compute HCF stepwise: gcd(132, 204) first, then gcd(result, 228). The Euclidean algorithm is efficient and reliable for such calculations.

Step-by-Step Solution:

Verification / Alternative check:
Prime factors: 132 = 2^2*3*11; 204 = 2^2*3*17; 228 = 2^2*3*19. The common prime product is 2^2*3 = 12, confirming the result.

Why Other Options Are Wrong:

• 18, 6, 21, 24: These either do not divide all three numbers or are not the greatest common factor. Only 12 divides all and is maximal.

Common Pitfalls:

• Selecting a common divisor (like 6) and overlooking a larger common divisor (12).

Final Answer:
12