Find the HCF of 306 and 657 by the prime factorisation method. Hence find their LCM.

Answer

To find the HCF (Highest Common Factor) of 306 and 657 using the prime factorization method and then find their LCM (Lowest Common Multiple), we can follow these steps:

### Step 1: Prime Factorization of the Numbers

**Prime Factorization of 306:**
1. Start by dividing 306 by the smallest prime number, which is 2:
\[ 306 \div 2 = 153 \]
So, \( 306 = 2 \times 153 \).

2. Next, factor 153. The smallest prime number that divides 153 is 3:
\[ 153 \div 3 = 51 \]
So, \( 153 = 3 \times 51 \).

3. Now, factor 51. Again, the smallest prime number that divides 51 is 3:
\[ 51 \div 3 = 17 \]
So, \( 51 = 3 \times 17 \).

4. Finally, 17 is a prime number.

Thus, the prime factorization of 306 is:
\[ 306 = 2^1 \times 3^2 \times 17^1 \]

**Prime Factorization of 657:**
1. Start by dividing 657 by the smallest prime number, which is 3:
\[ 657 \div 3 = 219 \]
So, \( 657 = 3 \times 219 \).

2. Next, factor 219. The smallest prime number that divides 219 is again 3:
\[ 219 \div 3 = 73 \]
So, \( 219 = 3 \times 73 \).

3. Finally, 73 is a prime number.

Thus, the prime factorization of 657 is:
\[ 657 = 3^2 \times 73^1 \]

### Step 2: Finding Common Factors

Now we identify the common prime factors from both factorizations:
- From \( 306 = 2^1 \times 3^2 \times 17^1 \)
- From \( 657 = 3^2 \times 73^1 \)

The common prime factor is:
- \( 3^2 \)

### Step 3: Calculate HCF

To find the HCF, we multiply the common factors:
\[ \text{HCF} = 3^2 = 9 \]

### Step 4: Calculate LCM

Using the relationship between HCF and LCM:
\[ \text{HCF} \times \text{LCM} = \text{Product of the numbers} \]
\[ 9 \times \text{LCM} = 306 \times 657 \]

Calculating the product:
\[ 306 \times 657 = 201342 \]

Now, we can find LCM:
\[ \text{LCM} = \frac{306 \times 657}{9} = \frac{201342}{9} = 22374 \]

### Final Answers
- HCF of 306 and 657 is **9**.
- LCM of 306 and 657 is **22374**.