H.C.F of 42, 63, and 140 is -

Options

A. 14
B. 9
C. 21
D. 7

Answer

To find the H.C.F (Highest Common Factor) of the numbers 42, 63, and 140, we will follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 42**:
- Divide by 2: \( 42 \div 2 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- So, the prime factorization of 42 is \( 2^1 \times 3^1 \times 7^1 \).

- **For 63**:
- Divide by 3: \( 63 \div 3 = 21 \)
- Divide 21 by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.
- So, the prime factorization of 63 is \( 3^2 \times 7^1 \).

- **For 140**:
- Divide by 2: \( 140 \div 2 = 70 \)
- Divide 70 by 2: \( 70 \div 2 = 35 \)
- Divide 35 by 5: \( 35 \div 5 = 7 \)
- 7 is a prime number.
- So, the prime factorization of 140 is \( 2^2 \times 5^1 \times 7^1 \).

### Step 2: Identify Common Factors
Next, we identify the common prime factors from the factorizations:
- The prime factors of 42 are \( 2^1, 3^1, 7^1 \).
- The prime factors of 63 are \( 3^2, 7^1 \).
- The prime factors of 140 are \( 2^2, 5^1, 7^1 \).

The common prime factor among all three numbers is \( 7^1 \).

### Step 3: Calculate H.C.F
The H.C.F is obtained by taking the lowest power of the common prime factors:
- For the common prime factor \( 7 \), the lowest power is \( 7^1 \).

Thus, the H.C.F of 42, 63, and 140 is \( 7 \).

### Final Answer
The H.C.F of 42, 63, and 140 is **7**.

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