Find the LCM and HCF of the of the following integers by applying the prime factorisation method.
12, 15, and 21

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of the integers 12, 15, and 21 using the prime factorization method, we will follow these steps:

### Step 1: Prime Factorization of Each Number
- **For 12**:
- Divide by 2: \( 12 \div 2 = 6 \)
- Divide by 2 again: \( 6 \div 2 = 3 \)
- Divide by 3: \( 3 \div 3 = 1 \)
- Thus, the prime factorization of 12 is \( 2^2 \times 3^1 \).

- **For 15**:
- Divide by 3: \( 15 \div 3 = 5 \)
- Divide by 5: \( 5 \div 5 = 1 \)
- Thus, the prime factorization of 15 is \( 3^1 \times 5^1 \).

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

### Step 2: Write the Prime Factorizations
- 12 = \( 2^2 \times 3^1 \)
- 15 = \( 3^1 \times 5^1 \)
- 21 = \( 3^1 \times 7^1 \)

### Step 3: Finding HCF
To find the HCF, we take the lowest power of all prime factors that are common to all three numbers:
- The only common prime factor is \( 3 \).
- The lowest power of \( 3 \) in all three factorizations is \( 3^1 \).

Thus, the HCF is:
\[ \text{HCF} = 3^1 = 3 \]

### Step 4: Finding LCM
To find the LCM, we take the highest power of all prime factors present in any of the numbers:
- For \( 2 \): highest power is \( 2^2 \) (from 12).
- For \( 3 \): highest power is \( 3^1 \) (common in all).
- For \( 5 \): highest power is \( 5^1 \) (from 15).
- For \( 7 \): 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:
\[ = 4 \times 3 \times 5 \times 7 \]
\[ = 12 \times 5 \times 7 \]
\[ = 60 \times 7 = 420 \]

### Final Answers
- HCF = 3
- LCM = 420

---