Find the HCF and LCM of the following by prime factorisation method :
(i) 12 and 15 (ii) 20 and 25 (iii) 28 and 42 (iv) 336 and 56 (v) 12,15 and 21 (vi)10,20 and 30

Answer

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

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

1. **Prime Factorization**:
- 12 = 2 × 2 × 3 = \(2^2 \times 3^1\)
- 15 = 3 × 5 = \(3^1 \times 5^1\)

2. **Finding HCF**:
- Common factors: 3
- HCF = \(3^1 = 3\)

3. **Finding LCM**:
- LCM = \(2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\)

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

1. **Prime Factorization**:
- 20 = 2 × 10 = 2 × 2 × 5 = \(2^2 \times 5^1\)
- 25 = 5 × 5 = \(5^2\)

2. **Finding HCF**:
- Common factors: 5
- HCF = \(5^1 = 5\)

3. **Finding LCM**:
- LCM = \(2^2 \times 5^2 = 4 \times 25 = 100\)

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

1. **Prime Factorization**:
- 28 = 2 × 14 = 2 × 2 × 7 = \(2^2 \times 7^1\)
- 42 = 2 × 21 = 2 × 3 × 7 = \(2^1 \times 3^1 \times 7^1\)

2. **Finding HCF**:
- Common factors: 2 and 7
- HCF = \(2^1 \times 7^1 = 14\)

3. **Finding LCM**:
- LCM = \(2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84\)

### (iv) HCF and LCM of 336 and 56

1. **Prime Factorization**:
- 336 = 2 × 168 = 2 × 84 = 2 × 42 = 2 × 21 = \(2^4 \times 3^1 \times 7^1\)
- 56 = 2 × 28 = 2 × 14 = 2 × 7 = \(2^3 \times 7^1\)

2. **Finding HCF**:
- Common factors: 2 and 7
- HCF = \(2^3 \times 7^1 = 56\)

3. **Finding LCM**:
- LCM = \(2^4 \times 3^1 \times 7^1 = 16 \times 3 \times 7 = 336\)

### (v) HCF and LCM of 12, 15, and 21

1. **Prime Factorization**:
- 12 = \(2^2 \times 3^1\)
- 15 = \(3^1 \times 5^1\)
- 21 = \(3^1 \times 7^1\)

2. **Finding HCF**:
- Common factor: 3
- HCF = \(3^1 = 3\)

3. **Finding LCM**:
- LCM = \(2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420\)

### (vi) HCF and LCM of 10, 20, and 30

1. **Prime Factorization**:
- 10 = \(2^1 \times 5^1\)
- 20 = \(2^2 \times 5^1\)
- 30 = \(2^1 \times 3^1 \times 5^1\)

2. **Finding HCF**:
- Common factors: 2 and 5
- HCF = \(2^1 \times 5^1 = 10\)

3. **Finding LCM**:
- LCM = \(2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\)

### Summary of Results

- (i) HCF = 3, LCM = 60
- (ii) HCF = 5, LCM = 100
- (iii) HCF = 14, LCM = 84
- (iv) HCF = 56, LCM = 336
- (v) HCF = 3, LCM = 420
- (vi) HCF = 10, LCM = 60