HCF of `168` and `126` is

Options

A. `21`
B. `42`
C. `14`
D. `18`

Explanation

To find the HCF (Highest Common Factor) of 168 and 126, we can follow these steps:

### Step 1: Prime Factorization of 126
First, we need to factorize 126 into its prime factors.

- Divide 126 by 2 (the smallest prime number):
\[ 126 \div 2 = 63 \]
- Now, factor 63:
- Divide 63 by 3:
\[ 63 \div 3 = 21 \]
- Divide 21 by 3:
\[ 21 \div 3 = 7 \]
- 7 is a prime number.

So, the prime factorization of 126 is:
\[ 126 = 2^1 \times 3^2 \times 7^1 \]

### Step 2: Prime Factorization of 168
Next, we factorize 168 into its prime factors.

- Divide 168 by 2:
\[ 168 \div 2 = 84 \]
- Divide 84 by 2:
\[ 84 \div 2 = 42 \]
- Divide 42 by 2:
\[ 42 \div 2 = 21 \]
- Now, factor 21:
- Divide 21 by 3:
\[ 21 \div 3 = 7 \]
- 7 is a prime number.

So, the prime factorization of 168 is:
\[ 168 = 2^3 \times 3^1 \times 7^1 \]

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

- For 2: The minimum power is \(2^1\) (from 126).
- For 3: The minimum power is \(3^1\) (from 168).
- For 7: The minimum power is \(7^1\) (common in both).

### Step 4: Calculate HCF
Now, we multiply the common prime factors together to find the HCF:
\[ HCF = 2^1 \times 3^1 \times 7^1 = 2 \times 3 \times 7 \]
Calculating this:
\[ 2 \times 3 = 6 \]
\[ 6 \times 7 = 42 \]

Thus, the HCF of 168 and 126 is:
\[ \text{HCF} = 42 \]

### Final Answer:
The HCF of 168 and 126 is **42**.

---