Find the HCF of 396 and 1080 by the prime factorization method.

Answer

To find the HCF (Highest Common Factor) of 396 and 1080 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of 396
To find the prime factors of 396, we can divide it by the smallest prime numbers:

1. **Divide by 2** (the smallest prime number):
- \( 396 \div 2 = 198 \)
2. **Divide 198 by 2**:
- \( 198 \div 2 = 99 \)
3. **Divide 99 by 3** (next smallest prime):
- \( 99 \div 3 = 33 \)
4. **Divide 33 by 3**:
- \( 33 \div 3 = 11 \)
5. **11 is a prime number**, so we stop here.

Thus, the prime factorization of 396 is:
\[ 396 = 2^2 \times 3^2 \times 11 \]

### Step 2: Prime Factorization of 1080
Now, we will find the prime factors of 1080:

1. **Divide by 2**:
- \( 1080 \div 2 = 540 \)
2. **Divide 540 by 2**:
- \( 540 \div 2 = 270 \)
3. **Divide 270 by 2**:
- \( 270 \div 2 = 135 \)
4. **Divide 135 by 3**:
- \( 135 \div 3 = 45 \)
5. **Divide 45 by 3**:
- \( 45 \div 3 = 15 \)
6. **Divide 15 by 3**:
- \( 15 \div 3 = 5 \)
7. **5 is a prime number**, so we stop here.

Thus, the prime factorization of 1080 is:
\[ 1080 = 2^3 \times 3^3 \times 5 \]

### Step 3: Identify Common Factors
Now we will identify the common prime factors from both factorizations:

- **For 396**: \( 2^2 \times 3^2 \times 11 \)
- **For 1080**: \( 2^3 \times 3^3 \times 5 \)

The common prime factors are:
- The minimum power of 2 is \( 2^2 \).
- The minimum power of 3 is \( 3^2 \).

### Step 4: Calculate the HCF
Now we multiply the common prime factors:
\[ HCF = 2^2 \times 3^2 \]

Calculating this:
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)

Now multiply these results:
\[ HCF = 4 \times 9 = 36 \]

### Final Answer
The HCF of 396 and 1080 is **36**.

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