Determine the HCF of 882 and 840 .

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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**
The HCF of 882 and 840 is 42.

Determine the HCF of 882 and 840 .

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