Find the HCF of 513, 1134 and 1215

Options

A. 27
B. 9
C. 18
D. 54

Answer

To find the HCF (Highest Common Factor) of the numbers 513, 1134, and 1215, we will use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of Each Number

**1. Factorizing 513:**
- Divide by 3: \( 513 \div 3 = 171 \)
- Divide by 3 again: \( 171 \div 3 = 57 \)
- Divide by 3 again: \( 57 \div 3 = 19 \)
- 19 is a prime number.
- Therefore, the prime factorization of 513 is:
\[ 513 = 3^3 \times 19 \]

**2. Factorizing 1134:**
- Divide by 2: \( 1134 \div 2 = 567 \)
- Divide by 3: \( 567 \div 3 = 189 \)
- Divide by 3 again: \( 189 \div 3 = 63 \)
- Divide by 3 again: \( 63 \div 3 = 21 \)
- Divide by 3 again: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- Therefore, the prime factorization of 1134 is:
\[ 1134 = 2^1 \times 3^4 \times 7^1 \]

**3. Factorizing 1215:**
- Divide by 3: \( 1215 \div 3 = 405 \)
- Divide by 3 again: \( 405 \div 3 = 135 \)
- Divide by 3 again: \( 135 \div 3 = 45 \)
- Divide by 3 again: \( 45 \div 3 = 15 \)
- Divide by 3 again: \( 15 \div 3 = 5 \)
- 5 is a prime number.
- Therefore, the prime factorization of 1215 is:
\[ 1215 = 3^5 \times 5^1 \]

### Step 2: Identify Common Factors

Now we will identify the common factors from the prime factorizations:
- **513:** \( 3^3 \times 19 \)
- **1134:** \( 2^1 \times 3^4 \times 7^1 \)
- **1215:** \( 3^5 \times 5^1 \)

The only common prime factor among all three numbers is \( 3 \).

### Step 3: Determine the Lowest Power of Common Factors

The lowest power of the common factor \( 3 \) is:
- From 513: \( 3^3 \)
- From 1134: \( 3^4 \)
- From 1215: \( 3^5 \)

The lowest power is \( 3^3 \).

### Step 4: Calculate the HCF

Now we calculate the HCF:
\[ \text{HCF} = 3^3 = 27 \]

### Conclusion

Thus, the HCF of 513, 1134, and 1215 is **27**.

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