H.C.F. of 84, 63 and 42 is

Options

A. 9
B. 21
C. 7
D. 42

Answer

To find the H.C.F. (Highest Common Factor) of the numbers 84, 63, and 42, we will follow these steps:

### Step 1: Prime Factorization
We will start by finding the prime factorization of each number.

1. **Prime Factorization of 84:**
- 84 can be divided by 2 (the smallest prime number):
\[ 84 \div 2 = 42 \]
- Next, divide 42 by 2 again:
\[ 42 \div 2 = 21 \]
- Now, divide 21 by 3 (the next smallest prime number):
\[ 21 \div 3 = 7 \]
- Finally, 7 is a prime number itself.
- Thus, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

2. **Prime Factorization of 63:**
- 63 can be divided by 3:
\[ 63 \div 3 = 21 \]
- Divide 21 by 3 again:
\[ 21 \div 3 = 7 \]
- 7 is a prime number.
- Thus, the prime factorization of 63 is:
\[ 63 = 3^2 \times 7^1 \]

3. **Prime Factorization of 42:**
- 42 can be divided by 2:
\[ 42 \div 2 = 21 \]
- Divide 21 by 3:
\[ 21 \div 3 = 7 \]
- 7 is a prime number.
- Thus, the prime factorization of 42 is:
\[ 42 = 2^1 \times 3^1 \times 7^1 \]

### Step 2: Identify Common Factors
Now, we will identify the common prime factors from the factorizations:

- For **2**:
- In 84: \(2^2\)
- In 63: \(2^0\) (not present)
- In 42: \(2^1\)
- The minimum power is \(2^0\).

- For **3**:
- In 84: \(3^1\)
- In 63: \(3^2\)
- In 42: \(3^1\)
- The minimum power is \(3^1\).

- For **7**:
- In 84: \(7^1\)
- In 63: \(7^1\)
- In 42: \(7^1\)
- The minimum power is \(7^1\).

### Step 3: Calculate the H.C.F.
Now we can calculate the H.C.F. by multiplying the common prime factors with their lowest powers:

\[ H.C.F. = 2^0 \times 3^1 \times 7^1 = 1 \times 3 \times 7 = 21 \]

Thus, the H.C.F. of 84, 63, and 42 is **21**.