The HCF and the LCM of 12,21,15 respectively are

Options

A. 3,140
B. 12,420
C. 3,420
D. 420,3

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 12, 21, and 15, we will use the prime factorization method. Here’s a step-by-step solution:

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

- **For 12**:
- 12 can be divided by 2: \( 12 \div 2 = 6 \)
- 6 can be divided by 2: \( 6 \div 2 = 3 \)
- 3 is a prime number.

So, the prime factorization of 12 is:
\[ 12 = 2^2 \times 3^1 \]

- **For 21**:
- 21 can be divided by 3: \( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 21 is:
\[ 21 = 3^1 \times 7^1 \]

- **For 15**:
- 15 can be divided by 3: \( 15 \div 3 = 5 \)
- 5 is a prime number.

So, the prime factorization of 15 is:
\[ 15 = 3^1 \times 5^1 \]

### Step 2: Finding the HCF
The HCF is found by taking the lowest power of all common prime factors.

- The common prime factor is \( 3 \).
- The lowest power of \( 3 \) in all three factorizations is \( 3^1 \).

Thus, the HCF is:
\[ \text{HCF} = 3 \]

### Step 3: Finding the LCM
The LCM is found by taking the highest power of all prime factors present in any of the numbers.

- For \( 2 \): The highest power is \( 2^2 \) (from 12).
- For \( 3 \): The highest power is \( 3^1 \) (common in all).
- For \( 5 \): The highest power is \( 5^1 \) (from 15).
- For \( 7 \): The highest power is \( 7^1 \) (from 21).

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 \]

Calculating this step-by-step:
1. \( 2^2 = 4 \)
2. \( 4 \times 3 = 12 \)
3. \( 12 \times 5 = 60 \)
4. \( 60 \times 7 = 420 \)

So, the LCM is:
\[ \text{LCM} = 420 \]

### Final Answer
- HCF of 12, 21, and 15 is **3**.
- LCM of 12, 21, and 15 is **420**.