Find the HCF and LCM of the following by prime factorisation method :
(i) 12 and 15 (ii) 20 and 25 (iii) 28 and 42

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the given pairs of numbers using the prime factorization method, we will follow these steps for each pair.

### (i) Finding HCF and LCM of 12 and 15

**Step 1: Prime Factorization**
- Prime factorization of 12:
- 12 = 2 × 2 × 3 = \(2^2 × 3^1\)
- Prime factorization of 15:
- 15 = 3 × 5 = \(3^1 × 5^1\)

**Step 2: Identify Common Factors for HCF**
- The common prime factor is 3.
- The minimum power of 3 is \(3^1\).
- Therefore, HCF = \(3^1 = 3\).

**Step 3: Identify All Factors for LCM**
- For LCM, we take the highest power of all prime factors:
- For 2: \(2^2\)
- For 3: \(3^1\)
- For 5: \(5^1\)
- Thus, LCM = \(2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60\).

### (ii) Finding HCF and LCM of 20 and 25

**Step 1: Prime Factorization**
- Prime factorization of 20:
- 20 = 2 × 2 × 5 = \(2^2 × 5^1\)
- Prime factorization of 25:
- 25 = 5 × 5 = \(5^2\)

**Step 2: Identify Common Factors for HCF**
- The common prime factor is 5.
- The minimum power of 5 is \(5^1\).
- Therefore, HCF = \(5^1 = 5\).

**Step 3: Identify All Factors for LCM**
- For LCM, we take the highest power of all prime factors:
- For 2: \(2^2\)
- For 5: \(5^2\)
- Thus, LCM = \(2^2 × 5^2 = 4 × 25 = 100\).

### (iii) Finding HCF and LCM of 28 and 42

**Step 1: Prime Factorization**
- Prime factorization of 28:
- 28 = 2 × 2 × 7 = \(2^2 × 7^1\)
- Prime factorization of 42:
- 42 = 2 × 3 × 7 = \(2^1 × 3^1 × 7^1\)

**Step 2: Identify Common Factors for HCF**
- The common prime factors are 2 and 7.
- The minimum power of 2 is \(2^1\) and for 7 is \(7^1\).
- Therefore, HCF = \(2^1 × 7^1 = 2 × 7 = 14\).

**Step 3: Identify All Factors for LCM**
- For LCM, we take the highest power of all prime factors:
- For 2: \(2^2\)
- For 3: \(3^1\)
- For 7: \(7^1\)
- Thus, LCM = \(2^2 × 3^1 × 7^1 = 4 × 3 × 7 = 84\).

### Final Results:
- (i) HCF = 3, LCM = 60
- (ii) HCF = 5, LCM = 100
- (iii) HCF = 14, LCM = 84