Find the HCF of he numbers in each of the following , using the prime factorization method :
`84, 120, 138`

Options

A. `12`
B. `6`
C. `18`
D. `3`

Answer

To find the HCF (Highest Common Factor) of the numbers 84, 120, and 138 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number

**Prime Factorization of 84:**
1. Divide 84 by the smallest prime number, which is 2:
- \( 84 \div 2 = 42 \)
2. Divide 42 by 2:
- \( 42 \div 2 = 21 \)
3. Divide 21 by the next smallest prime number, which is 3:
- \( 21 \div 3 = 7 \)
4. Finally, 7 is a prime number, so we stop here.

Thus, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

**Prime Factorization of 120:**
1. Divide 120 by 2:
- \( 120 \div 2 = 60 \)
2. Divide 60 by 2:
- \( 60 \div 2 = 30 \)
3. Divide 30 by 2:
- \( 30 \div 2 = 15 \)
4. Divide 15 by 3:
- \( 15 \div 3 = 5 \)
5. Finally, 5 is a prime number, so we stop here.

Thus, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

**Prime Factorization of 138:**
1. Divide 138 by 2:
- \( 138 \div 2 = 69 \)
2. Divide 69 by 3:
- \( 69 \div 3 = 23 \)
3. Finally, 23 is a prime number, so we stop here.

Thus, the prime factorization of 138 is:
\[ 138 = 2^1 \times 3^1 \times 23^1 \]

### Step 2: Identify Common Prime Factors

Now, we will identify the common prime factors from the factorizations:
- For 84: \( 2^2 \times 3^1 \times 7^1 \)
- For 120: \( 2^3 \times 3^1 \times 5^1 \)
- For 138: \( 2^1 \times 3^1 \times 23^1 \)

The common prime factors are:
- **2** (with the minimum power being \( 2^1 \))
- **3** (with the minimum power being \( 3^1 \))

### Step 3: Calculate the HCF

To find the HCF, we multiply the common prime factors raised to their minimum powers:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Final Answer
The HCF of 84, 120, and 138 is **6**.

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