Find the HCF of 42 and 105.

Answer

To find the Highest Common Factor (HCF) of 42 and 105, we will follow these steps:

### Step 1: Find the Prime Factorization of 42
- Start by dividing 42 by the smallest prime number, which is 2.
- \( 42 \div 2 = 21 \)
- Next, factor 21. The smallest prime number that divides 21 is 3.
- \( 21 \div 3 = 7 \)
- Finally, 7 is a prime number itself.
- So, the prime factorization of 42 is \( 2 \times 3 \times 7 \).

### Step 2: Find the Prime Factorization of 105
- Start by dividing 105 by the smallest prime number, which is 3 (since 105 is odd and not divisible by 2).
- \( 105 \div 3 = 35 \)
- Next, factor 35. The smallest prime number that divides 35 is 5.
- \( 35 \div 5 = 7 \)
- Finally, 7 is a prime number itself.
- So, the prime factorization of 105 is \( 3 \times 5 \times 7 \).

### Step 3: Identify the Common Prime Factors
- Now, we will list the prime factors of both numbers:
- Prime factors of 42: \( 2, 3, 7 \)
- Prime factors of 105: \( 3, 5, 7 \)
- The common prime factors are \( 3 \) and \( 7 \).

### Step 4: Calculate the HCF
- To find the HCF, multiply the common prime factors:
- \( HCF = 3 \times 7 = 21 \)

### Final Answer
- The HCF of 42 and 105 is **21**.

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