Find the HCF of 210 and 392. Hence, find their LCM.

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 210 and 392, we will use the prime factorization method.

### Step 1: Prime Factorization of 210
1. Start with the number 210.
2. Divide by the smallest prime number, which is 2:
- \( 210 \div 2 = 105 \)
3. Next, divide 105 by the smallest prime number that divides it, which is 3:
- \( 105 \div 3 = 35 \)
4. Now, divide 35 by the smallest prime number that divides it, which is 5:
- \( 35 \div 5 = 7 \)
5. Finally, 7 is a prime number, so we stop here.

The prime factorization of 210 is:
\[ 210 = 2^1 \times 3^1 \times 5^1 \times 7^1 \]

### Step 2: Prime Factorization of 392
1. Start with the number 392.
2. Divide by the smallest prime number, which is 2:
- \( 392 \div 2 = 196 \)
3. Divide 196 by 2 again:
- \( 196 \div 2 = 98 \)
4. Divide 98 by 2 again:
- \( 98 \div 2 = 49 \)
5. Now, divide 49 by the smallest prime number that divides it, which is 7:
- \( 49 \div 7 = 7 \)
6. Finally, 7 is a prime number, so we stop here.

The prime factorization of 392 is:
\[ 392 = 2^3 \times 7^2 \]

### Step 3: Finding the HCF
1. Identify the common prime factors from both factorizations:
- For 2: The minimum power is \( 2^1 \)
- For 7: The minimum power is \( 7^1 \)
2. Multiply the common prime factors:
\[ \text{HCF} = 2^1 \times 7^1 = 2 \times 7 = 14 \]

### Step 4: Finding the LCM
1. Use the formula:
\[ \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} \]
2. Substitute the values:
\[ \text{LCM}(210, 392) = \frac{210 \times 392}{14} \]
3. Calculate \( 210 \times 392 \):
- \( 210 \times 392 = 82320 \)
4. Divide by HCF:
\[ \text{LCM} = \frac{82320}{14} = 5880 \]

### Final Answer
- HCF of 210 and 392 is **14**.
- LCM of 210 and 392 is **5880**.