Find the HCF of `12` and `56` by prime factorization method.

Options

A. `2`
B. `4`
C. `3`
D. `8`

Explanation

To find the HCF (Highest Common Factor) of 12 and 56 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 12
- Start by dividing 12 by the smallest prime number, which is 2.
- \( 12 \div 2 = 6 \)
- Next, divide 6 by 2 again.
- \( 6 \div 2 = 3 \)
- Now, 3 is a prime number, so we stop here.
- Therefore, the prime factorization of 12 is:
\[ 12 = 2^2 \times 3^1 \]

### Step 2: Prime Factorization of 56
- Start by dividing 56 by the smallest prime number, which is 2.
- \( 56 \div 2 = 28 \)
- Divide 28 by 2 again.
- \( 28 \div 2 = 14 \)
- Divide 14 by 2 again.
- \( 14 \div 2 = 7 \)
- Now, 7 is a prime number, so we stop here.
- Therefore, the prime factorization of 56 is:
\[ 56 = 2^3 \times 7^1 \]

### Step 3: Identify Common Factors
- Now, we need to identify the common prime factors from the factorizations of 12 and 56.
- The prime factors of 12 are \( 2^2 \) and \( 3^1 \).
- The prime factors of 56 are \( 2^3 \) and \( 7^1 \).
- The common prime factor is \( 2 \).

### Step 4: Determine the Lowest Power of Common Factors
- For the common factor \( 2 \):
- The power of 2 in 12 is \( 2^2 \).
- The power of 2 in 56 is \( 2^3 \).
- The lowest power of the common factor \( 2 \) is \( 2^2 \).

### Step 5: Calculate the HCF
- Therefore, the HCF is:
\[ HCF = 2^2 = 4 \]

### Final Answer
The HCF of 12 and 56 is **4**.

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