Find the highest common factor of $36$ and $84$.

### Concept & Strategy For relatively small numbers, the Highest Common Factor (HCF) can be found using the difference method. The HCF of two numbers will always evenly divide their difference. This narrows down the possibilities significantly. Alternatively, basic prime factorization or looking for the largest common divisor from the options works well. ### Step-by-Step Solution * **Given:** The numbers are $36$ and $84$. * **Calculation / Deduction:** 1. **Find the Difference:** $$84 - 36 = 48$$ 2. The HCF must be a factor of $48$. The factors of $48$ are $1, 2, 3, 4, 6, 8, 12, 16, 24, 48$. 3. Now, we check these factors against our given options, starting from the largest option to find the *highest* common factor. 4. Let's test Option (d) $18$: $18$ is not a factor of $48$. (Skip) 5. Let's test Option (c) $12$: $12$ is a factor of $48$. Let's see if it divides both $36$ and $84$: $$36 \div 12 = 3$$ $$84 \div 12 = 7$$ 6. Since $12$ divides both perfectly, and we are working downwards from the largest options, $12$ is the HCF. ### Exam Strategy & Shortcut **Option Testing from Largest to Smallest:** Since you are looking for the *Highest* Common Factor, start testing the given options from largest to smallest. 1. Test 18: $36$ is divisible by $18$, but $84 \div 18$ is not a whole number. 2. Test 12: $36 \div 12 = 3$. $84 \div 12 = 7$. Both divide cleanly. You have found your HCF. This bypasses all manual calculation and is incredibly fast for two-digit numbers. ### Common Pitfall Starting the option testing from the smallest number. If you check $4$ or $6$ first, you will find they *do* divide both $36$ and $84$. If you stop there, you will select a common factor, but not the *highest* common factor. Always test in descending order. ### Final Answer **Therefore, the correct answer is $12$.**