Reduce `(264)/(408)` to simplest form using HCF of the numerator and denominator

Answer

To reduce the fraction \( \frac{264}{408} \) to its simplest form using the Highest Common Factor (HCF), we will follow these steps:

### Step-by-Step Solution:

1. **Find the Prime Factorization of 264**:
- Start dividing 264 by the smallest prime number, which is 2.
- \( 264 \div 2 = 132 \)
- \( 132 \div 2 = 66 \)
- \( 66 \div 2 = 33 \)
- \( 33 \div 3 = 11 \)
- Since 11 is a prime number, we stop here.
- Therefore, the prime factorization of 264 is:
\[ 264 = 2^3 \times 3 \times 11 \]

2. **Find the Prime Factorization of 408**:
- Start dividing 408 by the smallest prime number, which is 2.
- \( 408 \div 2 = 204 \)
- \( 204 \div 2 = 102 \)
- \( 102 \div 2 = 51 \)
- \( 51 \div 3 = 17 \)
- Since 17 is a prime number, we stop here.
- Therefore, the prime factorization of 408 is:
\[ 408 = 2^3 \times 3 \times 17 \]

3. **Identify the HCF**:
- The HCF is found by taking the lowest power of all common prime factors.
- The common prime factors of 264 and 408 are \( 2^3 \) and \( 3 \).
- Thus, the HCF is:
\[ HCF = 2^3 \times 3 = 8 \times 3 = 24 \]

4. **Divide the Numerator and Denominator by the HCF**:
- Divide the numerator (264) by the HCF (24):
\[ \frac{264}{24} = 11 \]
- Divide the denominator (408) by the HCF (24):
\[ \frac{408}{24} = 17 \]

5. **Write the Simplified Fraction**:
- The fraction \( \frac{264}{408} \) in its simplest form is:
\[ \frac{11}{17} \]

### Final Answer:
The simplest form of \( \frac{264}{408} \) is \( \frac{11}{17} \).

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