Using prime factorisation method, find the HCF and LCM of the following pairs. Hence, verify `HCF xx LCM`= product of two numbers. (i) 96 and 120 (ii) 16 and 20 (iii) 144 and 192 (iv) 330 and 440

Answer

To find the HCF and LCM of the given pairs of numbers using the prime factorization method, we will follow these steps for each pair:

### (i) For 96 and 120:

**Step 1: Prime Factorization**
- **96**:
- Divide by 2: 96 ÷ 2 = 48
- Divide by 2: 48 ÷ 2 = 24
- Divide by 2: 24 ÷ 2 = 12
- Divide by 2: 12 ÷ 2 = 6
- Divide by 2: 6 ÷ 2 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 96 is \( 2^5 \times 3^1 \).

- **120**:
- Divide by 2: 120 ÷ 2 = 60
- Divide by 2: 60 ÷ 2 = 30
- Divide by 2: 30 ÷ 2 = 15
- Divide by 3: 15 ÷ 3 = 5
- Divide by 5: 5 ÷ 5 = 1
- So, the prime factorization of 120 is \( 2^3 \times 3^1 \times 5^1 \).

**Step 2: Finding HCF**
- Take the lowest power of common prime factors:
- For 2: \( \min(5, 3) = 3 \) → \( 2^3 \)
- For 3: \( \min(1, 1) = 1 \) → \( 3^1 \)
- HCF = \( 2^3 \times 3^1 = 8 \times 3 = 24 \)

**Step 3: Finding LCM**
- Take the highest power of all prime factors:
- For 2: \( \max(5, 3) = 5 \) → \( 2^5 \)
- For 3: \( \max(1, 1) = 1 \) → \( 3^1 \)
- For 5: \( \max(0, 1) = 1 \) → \( 5^1 \)
- LCM = \( 2^5 \times 3^1 \times 5^1 = 32 \times 3 \times 5 = 480 \)

**Step 4: Verification**
- Product of the two numbers: \( 96 \times 120 = 11520 \)
- HCF × LCM: \( 24 \times 480 = 11520 \)
- Hence, \( HCF \times LCM = \) Product of the two numbers.

### (ii) For 16 and 20:

**Step 1: Prime Factorization**
- **16**: \( 2^4 \)
- **20**:
- Divide by 2: 20 ÷ 2 = 10
- Divide by 2: 10 ÷ 2 = 5
- Divide by 5: 5 ÷ 5 = 1
- So, the prime factorization of 20 is \( 2^2 \times 5^1 \).

**Step 2: Finding HCF**
- For 2: \( \min(4, 2) = 2 \) → \( 2^2 \)
- HCF = \( 2^2 = 4 \)

**Step 3: Finding LCM**
- For 2: \( \max(4, 2) = 4 \) → \( 2^4 \)
- For 5: \( \max(0, 1) = 1 \) → \( 5^1 \)
- LCM = \( 2^4 \times 5^1 = 16 \times 5 = 80 \)

**Step 4: Verification**
- Product of the two numbers: \( 16 \times 20 = 320 \)
- HCF × LCM: \( 4 \times 80 = 320 \)
- Hence, \( HCF \times LCM = \) Product of the two numbers.

### (iii) For 144 and 192:

**Step 1: Prime Factorization**
- **144**:
- Divide by 2: 144 ÷ 2 = 72
- Divide by 2: 72 ÷ 2 = 36
- Divide by 2: 36 ÷ 2 = 18
- Divide by 2: 18 ÷ 2 = 9
- Divide by 3: 9 ÷ 3 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 144 is \( 2^4 \times 3^2 \).

- **192**:
- Divide by 2: 192 ÷ 2 = 96
- Divide by 2: 96 ÷ 2 = 48
- Divide by 2: 48 ÷ 2 = 24
- Divide by 2: 24 ÷ 2 = 12
- Divide by 2: 12 ÷ 2 = 6
- Divide by 2: 6 ÷ 2 = 3
- Divide by 3: 3 ÷ 3 = 1
- So, the prime factorization of 192 is \( 2^6 \times 3^1 \).

**Step 2: Finding HCF**
- For 2: \( \min(4, 6) = 4 \) → \( 2^4 \)
- For 3: \( \min(2, 1) = 1 \) → \( 3^1 \)
- HCF = \( 2^4 \times 3^1 = 16 \times 3 = 48 \)

**Step 3: Finding LCM**
- For 2: \( \max(4, 6) = 6 \) → \( 2^6 \)
- For 3: \( \max(2, 1) = 2 \) → \( 3^2 \)
- LCM = \( 2^6 \times 3^2 = 64 \times 9 = 576 \)

**Step 4: Verification**
- Product of the two numbers: \( 144 \times 192 = 27648 \)
- HCF × LCM: \( 48 \times 576 = 27648 \)
- Hence, \( HCF \times LCM = \) Product of the two numbers.

### (iv) For 330 and 440:

**Step 1: Prime Factorization**
- **330**:
- Divide by 2: 330 ÷ 2 = 165
- Divide by 3: 165 ÷ 3 = 55
- Divide by 5: 55 ÷ 5 = 11
- Divide by 11: 11 ÷ 11 = 1
- So, the prime factorization of 330 is \( 2^1 \times 3^1 \times 5^1 \times 11^1 \).

- **440**:
- Divide by 2: 440 ÷ 2 = 220
- Divide by 2: 220 ÷ 2 = 110
- Divide by 2: 110 ÷ 2 = 55
- Divide by 5: 55 ÷ 5 = 11
- Divide by 11: 11 ÷ 11 = 1
- So, the prime factorization of 440 is \( 2^3 \times 5^1 \times 11^1 \).

**Step 2: Finding HCF**
- For 2: \( \min(1, 3) = 1 \) → \( 2^1 \)
- For 3: \( \min(1, 0) = 0 \) → (not included)
- For 5: \( \min(1, 1) = 1 \) → \( 5^1 \)
- For 11: \( \min(1, 1) = 1 \) → \( 11^1 \)
- HCF = \( 2^1 \times 5^1 \times 11^1 = 2 \times 5 \times 11 = 110 \)

**Step 3: Finding LCM**
- For 2: \( \max(1, 3) = 3 \) → \( 2^3 \)
- For 3: \( \max(1, 0) = 1 \) → \( 3^1 \)
- For 5: \( \max(1, 1) = 1 \) → \( 5^1 \)
- For 11: \( \max(1, 1) = 1 \) → \( 11^1 \)
- LCM = \( 2^3 \times 3^1 \times 5^1 \times 11^1 = 8 \times 3 \times 5 \times 11 = 1320 \)

**Step 4: Verification**
- Product of the two numbers: \( 330 \times 440 = 145200 \)
- HCF × LCM: \( 110 \times 1320 = 145200 \)
- Hence, \( HCF \times LCM = \) Product of the two numbers.

### Summary of Results:
- (i) HCF = 24, LCM = 480
- (ii) HCF = 4, LCM = 80
- (iii) HCF = 48, LCM = 576
- (iv) HCF = 110, LCM = 1320