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

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:

### Part (i): Find HCF and LCM of 96 and 120

**Step 1: Prime Factorization of 96**
- Divide 96 by 2:
- 96 ÷ 2 = 48
- Divide 48 by 2:
- 48 ÷ 2 = 24
- Divide 24 by 2:
- 24 ÷ 2 = 12
- Divide 12 by 2:
- 12 ÷ 2 = 6
- Divide 6 by 2:
- 6 ÷ 2 = 3
- Finally, divide 3 by 3:
- 3 ÷ 3 = 1

Thus, the prime factorization of 96 is:
\[ 96 = 2^5 \times 3^1 \]

**Step 2: Prime Factorization of 120**
- Divide 120 by 2:
- 120 ÷ 2 = 60
- Divide 60 by 2:
- 60 ÷ 2 = 30
- Divide 30 by 2:
- 30 ÷ 2 = 15
- Divide 15 by 3:
- 15 ÷ 3 = 5
- Finally, divide 5 by 5:
- 5 ÷ 5 = 1

Thus, the prime factorization of 120 is:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

**Step 3: Finding HCF**
- The HCF is found by taking the lowest power of all common prime factors.
- Common prime factors of 96 and 120 are 2 and 3.
- For 2: minimum power is \(2^3\)
- For 3: minimum power is \(3^1\)

Thus, the HCF is:
\[ HCF = 2^3 \times 3^1 = 8 \times 3 = 24 \]

**Step 4: Finding LCM**
- The LCM is found by taking the highest power of all prime factors present in either number.
- For 2: maximum power is \(2^5\)
- For 3: maximum power is \(3^1\)
- For 5: maximum power is \(5^1\)

Thus, the LCM is:
\[ LCM = 2^5 \times 3^1 \times 5^1 = 32 \times 3 \times 5 = 480 \]

**Step 5: Verification**
- We verify that \( HCF \times LCM = 96 \times 120 \)
- Calculate \( HCF \times LCM = 24 \times 480 = 11520 \)
- Calculate \( 96 \times 120 = 11520 \)

Since both products are equal, the verification holds true.

### Part (ii): Find HCF and LCM of 16 and 20

**Step 1: Prime Factorization of 16**
- Divide 16 by 2:
- 16 ÷ 2 = 8
- Divide 8 by 2:
- 8 ÷ 2 = 4
- Divide 4 by 2:
- 4 ÷ 2 = 2
- Finally, divide 2 by 2:
- 2 ÷ 2 = 1

Thus, the prime factorization of 16 is:
\[ 16 = 2^4 \]

**Step 2: Prime Factorization of 20**
- Divide 20 by 2:
- 20 ÷ 2 = 10
- Divide 10 by 2:
- 10 ÷ 2 = 5
- Finally, divide 5 by 5:
- 5 ÷ 5 = 1

Thus, the prime factorization of 20 is:
\[ 20 = 2^2 \times 5^1 \]

**Step 3: Finding HCF**
- The HCF is found by taking the lowest power of all common prime factors.
- Common prime factor is 2.
- For 2: minimum power is \(2^2\)

Thus, the HCF is:
\[ HCF = 2^2 = 4 \]

**Step 4: Finding LCM**
- The LCM is found by taking the highest power of all prime factors present in either number.
- For 2: maximum power is \(2^4\)
- For 5: maximum power is \(5^1\)

Thus, the LCM is:
\[ LCM = 2^4 \times 5^1 = 16 \times 5 = 80 \]

**Step 5: Verification**
- We verify that \( HCF \times LCM = 16 \times 20 \)
- Calculate \( HCF \times LCM = 4 \times 80 = 320 \)
- Calculate \( 16 \times 20 = 320 \)

Since both products are equal, the verification holds true.

### Final Results
- For the pair (96, 120):
- HCF = 24
- LCM = 480
- For the pair (16, 20):
- HCF = 4
- LCM = 80