Find the H.C.F. of "` 12, 18, 46

Options

A. `2`
B. `4`
C. `6`
D. None of these

Answer

To find the H.C.F. (Highest Common Factor) of the numbers 12, 18, and 46, we will follow these steps:

### Step 1: Factorize each number
1. **Factorize 12**:
- 12 is divisible by 2: \( 12 \div 2 = 6 \)
- 6 is also divisible by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 12 is:
\[ 12 = 2 \times 2 \times 3 = 2^2 \times 3^1 \]

2. **Factorize 18**:
- 18 is divisible by 2: \( 18 \div 2 = 9 \)
- 9 is divisible by 3: \( 9 \div 3 = 3 \)
- 3 is a prime number.
- Therefore, the prime factorization of 18 is:
\[ 18 = 2^1 \times 3^2 \]

3. **Factorize 46**:
- 46 is divisible by 2: \( 46 \div 2 = 23 \)
- 23 is a prime number.
- Therefore, the prime factorization of 46 is:
\[ 46 = 2^1 \times 23^1 \]

### Step 2: Identify common factors
Now we will identify the common factors from the factorizations:
- **Factors of 12**: \( 2^2, 3^1 \)
- **Factors of 18**: \( 2^1, 3^2 \)
- **Factors of 46**: \( 2^1, 23^1 \)

The only common factor among all three numbers is \( 2 \).

### Step 3: Determine the H.C.F.
The H.C.F. is the product of the lowest powers of all common prime factors:
- The only common prime factor is \( 2 \) with the lowest power of \( 2^1 \).

Thus, the H.C.F. of 12, 18, and 46 is:
\[ \text{H.C.F.} = 2^1 = 2 \]

### Final Answer
The H.C.F. of 12, 18, and 46 is **2**.

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