Find the HCF of 30, 42 and 135.

Answer

To find the HCF (Highest Common Factor) of the numbers 30, 42, and 135, we can follow these steps:

### Step 1: Find the Prime Factorization of Each Number
1. **Prime Factorization of 30**:
- Divide 30 by 2 (the smallest prime number):
- \( 30 \div 2 = 15 \)
- Divide 15 by 3 (the next smallest prime number):
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number itself.
- So, the prime factorization of 30 is:
\[ 30 = 2^1 \times 3^1 \times 5^1 \]

2. **Prime Factorization of 42**:
- Divide 42 by 2:
- \( 42 \div 2 = 21 \)
- Divide 21 by 3:
- \( 21 \div 3 = 7 \)
- Finally, 7 is a prime number.
- So, the prime factorization of 42 is:
\[ 42 = 2^1 \times 3^1 \times 7^1 \]

3. **Prime Factorization of 135**:
- Divide 135 by 3:
- \( 135 \div 3 = 45 \)
- Divide 45 by 3:
- \( 45 \div 3 = 15 \)
- Divide 15 by 3:
- \( 15 \div 3 = 5 \)
- Finally, 5 is a prime number.
- So, the prime factorization of 135 is:
\[ 135 = 3^3 \times 5^1 \]

### Step 2: Identify Common Prime Factors
Now, we list the prime factors of each number:
- **30**: \( 2^1, 3^1, 5^1 \)
- **42**: \( 2^1, 3^1, 7^1 \)
- **135**: \( 3^3, 5^1 \)

The 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:
- For 3, the lowest power in all three factorizations is \( 3^1 \).

### Step 4: Calculate the HCF
Thus, the HCF of 30, 42, and 135 is:
\[ HCF = 3^1 = 3 \]

### Final Answer
The HCF of 30, 42, and 135 is **3**.

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