Find the HCF of 132, 204 and 228.

### Concept & Strategy The Highest Common Factor (HCF) of a set of numbers is the largest number that divides all of them without leaving a remainder. The fastest way to find the HCF of large numbers is using the **Difference Method**: The HCF of any two numbers must also be a factor of their difference. $$\text{HCF}(a, b) \text{ must divide } |a - b|$$ ### Step-by-Step Solution **Given:** * Numbers: 132, 204, and 228. **Calculation:** 1. Find the differences between the adjacent numbers: * Difference 1: $204 - 132 = 72$ * Difference 2: $228 - 204 = 24$ 2. The HCF of the three numbers must be a factor of the smallest difference, which is $24$. 3. Look at the given options to find factors of 24: * (a) 12 - Is a factor of 24. * (b) 18 - Not a factor of 24. * (c) 6 - Is a factor of 24. * (d) 21 - Not a factor of 24. 4. Test the highest possible candidate first, which is 12: * $132 \div 12 = 11$ (Divisible) * $204 \div 12 = 17$ (Divisible) * $228 \div 12 = 19$ (Divisible) 5. Since 11, 17, and 19 are prime to each other (co-prime), 12 is indeed the highest common factor. ### Exam Strategy & Shortcut Use the difference method combined with option elimination. The difference between 228 and 204 is exactly 24. The HCF *cannot* be larger than 24, and it *must* divide 24 perfectly. Looking at the options (12, 18, 6, 21), 18 and 21 are instantly eliminated. Between 12 and 6, always test the larger number first since you are looking for the *Highest* Common Factor. ### Common Pitfall Resorting to full prime factorization for all three numbers ($132 = 2^2 \times 3 \times 11$, etc.) takes too much time under exam conditions and increases the chance of minor calculation errors. Use the difference method to save time. ### Final Answer **Therefore, the correct answer is 12.**