Find the LCM and HCF of 6 and 20 by the prime factorisation method.

Answer

To find the LCM (Least Common Multiple) and HCF (Highest Common Factor) of 6 and 20 using the prime factorization method, follow these steps:

### Step 1: Prime Factorization of 6
- Divide 6 by the smallest prime number, which is 2:
\[ 6 \div 2 = 3 \]
- Now, 3 is a prime number, so we stop here.
- Thus, the prime factorization of 6 is:
\[ 6 = 2^1 \times 3^1 \]

### Step 2: Prime Factorization of 20
- Divide 20 by the smallest prime number, which is 2:
\[ 20 \div 2 = 10 \]
- Divide 10 by 2 again:
\[ 10 \div 2 = 5 \]
- Now, 5 is a prime number, so we stop here.
- Thus, the prime factorization of 20 is:
\[ 20 = 2^2 \times 5^1 \]

### Step 3: Finding the HCF
- To find the HCF, we take the lowest power of all common prime factors.
- The common prime factor between 6 and 20 is 2.
- The lowest power of 2 in both factorizations is \(2^1\).
- Therefore, the HCF is:
\[ HCF = 2^1 = 2 \]

### Step 4: Finding the LCM
- To find the LCM, we take the highest power of all prime factors present in either number.
- The prime factors are 2, 3, and 5.
- The highest powers are:
- For 2: \(2^2\) (from 20)
- For 3: \(3^1\) (from 6)
- For 5: \(5^1\) (from 20)
- Therefore, the LCM is:
\[ LCM = 2^2 \times 3^1 \times 5^1 \]
- Calculating this gives:
\[ LCM = 4 \times 3 \times 5 = 12 \times 5 = 60 \]

### Final Results
- HCF of 6 and 20 is \(2\).
- LCM of 6 and 20 is \(60\).