Find the H.C.F. of 42, 63 and 140.

### Concept & Strategy To find the Highest Common Factor (H.C.F.) of small integers, the prime factorization method is highly effective. You break each number down into its core prime building blocks, then find the intersection (the common prime factors) and multiply their lowest powers. ### Step-by-Step Solution **Step 1: Perform prime factorization for each number.** * 42 = 2 $\times$ 3 $\times$ 7 * 63 = 3 $\times$ 3 $\times$ 7 = $3^2 \times 7$ * 140 = 2 $\times$ 2 $\times$ 5 $\times$ 7 = $2^2 \times 5 \times 7$ **Step 2: Identify the common prime factors.** * Looking at the factors, 2 is in 42 and 140, but not 63. * 3 is in 42 and 63, but not 140. * 5 is only in 140. * 7 is present in 42, 63, and 140. The only common prime factor is 7. **Step 3: Determine the lowest power.** The lowest power of 7 across all factorizations is $7^1$. ### Exam Strategy & Shortcut Use the **Difference Method** for speed. The H.C.F. of any set of numbers can never be larger than the smallest difference between any two of those numbers. Difference between 42 and 63 is 21. Therefore, the H.C.F. must be 21 or a factor of 21 (like 7 or 3). Check 21: 140 is not divisible by 21. Check 7: 42, 63, and 140 are all perfectly divisible by 7. The H.C.F. is 7. ### Common Pitfall Students often waste time doing long division for relatively small numbers like these. Long division is prone to arithmetic errors and takes longer; use the difference method or quick prime factorization instead. ### Final Answer **Therefore, the correct answer is 7.**