Determine the HCF of 882 and 840 .

To determine the Highest Common Factor (HCF) of **882** and **840**, we can use the **Euclidean Algorithm**, which is an efficient method for finding the greatest common divisor (GCD) of two numbers. Here's a step-by-step breakdown: ### **Step 1: Apply the Euclidean Algorithm** 1. **Divide 882 by 840**: \[ 882 \div 840 = 1 \text{ with a remainder of } 42 \] So, we can express this as: \[ 882 = 840 \times 1 + 42 \] 2. **Now, take 840 and divide it by the remainder (42)**: \[ 840 \div 42 = 20 \text{ with a remainder of } 0 \] This can be written as: \[ 840 = 42 \times 20 + 0 \] ### **Step 2: Determine the HCF** Since the remainder is now **0**, the divisor at this step (**42**) is the Highest Common Factor of **882** and **840**. ### **Verification via Prime Factorization** For additional verification, let's factorize both numbers into their prime factors. 1. **Prime Factorization of 882**: \[ 882 = 2 \times 3^2 \times 7^2 \] 2. **Prime Factorization of 840**: \[ 840 = 2^3 \times 3 \times 5 \times 7 \] 3. **Identify Common Prime Factors**: - **2** (minimum power: \(2^1\)) - **3** (minimum power: \(3^1\)) - **7** (minimum power: \(7^1\)) 4. **Multiply the Common Prime Factors**: \[ HCF = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 = 42 \] ### **Conclusion** The Highest Common Factor (HCF) of **882** and **840** is: **42**