Using prime factorisation, find the HCF and LCM of : , (i) 36, 84 (ii) 23,31 (iii) 96,404 (iv) 144,198 (v) 396, 1080 (vi) 1152, 1664 In each case, verify that : HCF ` xx` LCM = product of given number .

Answer

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

### (i) For 36 and 84:

1. **Prime Factorization of 36:**
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Thus, 36 = 2² × 3²

2. **Prime Factorization of 84:**
- 84 = 2 × 42
- 42 = 2 × 21
- 21 = 3 × 7
- Thus, 84 = 2² × 3 × 7

3. **Finding HCF:**
- Common factors: 2² and 3
- HCF = 2² × 3 = 4 × 3 = 12

4. **Finding LCM:**
- Combine all prime factors, taking the highest power of each:
- LCM = 2² × 3² × 7 = 4 × 9 × 7 = 252

5. **Verification:**
- HCF × LCM = 12 × 252 = 3024
- Product of 36 and 84 = 36 × 84 = 3024

### (ii) For 23 and 31:

1. **Prime Factorization of 23:**
- 23 is a prime number, so it is 23 = 1 × 23

2. **Prime Factorization of 31:**
- 31 is also a prime number, so it is 31 = 1 × 31

3. **Finding HCF:**
- HCF = 1 (since there are no common factors)

4. **Finding LCM:**
- LCM = 23 × 31 = 713

5. **Verification:**
- HCF × LCM = 1 × 713 = 713
- Product of 23 and 31 = 23 × 31 = 713

### (iii) For 96 and 404:

1. **Prime Factorization of 96:**
- 96 = 2 × 48
- 48 = 2 × 24
- 24 = 2 × 12
- 12 = 2 × 6
- 6 = 2 × 3
- Thus, 96 = 2⁵ × 3

2. **Prime Factorization of 404:**
- 404 = 2 × 202
- 202 = 2 × 101
- Thus, 404 = 2² × 101

3. **Finding HCF:**
- Common factors: 2²
- HCF = 2² = 4

4. **Finding LCM:**
- LCM = 2⁵ × 3 × 101 = 4 × 3 × 32 = 1212

5. **Verification:**
- HCF × LCM = 4 × 1212 = 4848
- Product of 96 and 404 = 96 × 404 = 4848

### (iv) For 144 and 198:

1. **Prime Factorization of 144:**
- 144 = 2 × 72
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Thus, 144 = 2⁴ × 3²

2. **Prime Factorization of 198:**
- 198 = 2 × 99
- 99 = 3 × 33
- 33 = 3 × 11
- Thus, 198 = 2 × 3² × 11

3. **Finding HCF:**
- Common factors: 2 and 3²
- HCF = 2 × 3² = 18

4. **Finding LCM:**
- LCM = 2⁴ × 3² × 11 = 792

5. **Verification:**
- HCF × LCM = 18 × 792 = 14256
- Product of 144 and 198 = 14256

### (v) For 396 and 1080:

1. **Prime Factorization of 396:**
- 396 = 2 × 198
- 198 = 2 × 99
- 99 = 3 × 33
- 33 = 3 × 11
- Thus, 396 = 2² × 3² × 11

2. **Prime Factorization of 1080:**
- 1080 = 2 × 540
- 540 = 2 × 270
- 270 = 2 × 135
- 135 = 3 × 45
- 45 = 3 × 15
- 15 = 3 × 5
- Thus, 1080 = 2³ × 3³ × 5

3. **Finding HCF:**
- Common factors: 2² and 3²
- HCF = 2² × 3² = 36

4. **Finding LCM:**
- LCM = 2³ × 3³ × 5 × 11 = 11880

5. **Verification:**
- HCF × LCM = 36 × 11880 = 427680
- Product of 396 and 1080 = 427680

### (vi) For 1152 and 1664:

1. **Prime Factorization of 1152:**
- 1152 = 2 × 576
- 576 = 2 × 288
- 288 = 2 × 144
- 144 = 2 × 72
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- Thus, 1152 = 2⁷ × 3²

2. **Prime Factorization of 1664:**
- 1664 = 2 × 832
- 832 = 2 × 416
- 416 = 2 × 208
- 208 = 2 × 104
- 104 = 2 × 52
- 52 = 2 × 26
- 26 = 2 × 13
- Thus, 1664 = 2⁶ × 13

3. **Finding HCF:**
- Common factors: 2⁶
- HCF = 2⁶ = 64

4. **Finding LCM:**
- LCM = 2⁷ × 3² × 13 = 4032

5. **Verification:**
- HCF × LCM = 64 × 4032 = 257408
- Product of 1152 and 1664 = 257408