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Example 4: Find the HCF and LCM of 6, 72, and 120 using the prime factorization method.

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GinnyAnsweridn
Sure! Let's find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 6, 72, and 120 using the prime factorisation method.

### Prime Factorization:

1. Prime Factorization of 6:
- 6 = 2 × 3

2. Prime Factorization of 72:
- 72 = 2 × 36
- 36 = 2 × 18
- 18 = 2 × 9
- 9 = 3 × 3
- So, 72 = 2² × 3²

3. Prime Factorization of 120:
- 120 = 2 × 60
- 60 = 2 × 30
- 30 = 2 × 15
- 15 = 3 × 5
- So, 120 = 2³ × 3 × 5

### Finding the HCF:

To find the HCF, we need to determine the common prime factors with the lowest powers for each of the numbers.

- Common prime factors: 2, 3
- Lowest power of 2 in the prime factorizations: 2¹ (appears in all numbers)
- Lowest power of 3 in the prime factorizations: 3¹ (appears in all numbers)

So, the HCF = 2¹ × 3¹ = 2 × 3 = 6

### Finding the LCM:

To find the LCM, we need to take each prime factor that appears in any of the numbers with the highest power.

- Prime factor: 2
- Highest power: 2³ (from 120)
- Prime factor: 3
- Highest power: 3² (from 72)
- Prime factor: 5
- Highest power: 5¹ (from 120)

So, the LCM = 2³ × 3² × 5¹ = 8 × 9 × 5 = 360.

### Summary:

- HCF of 6, 72, and 120: 6
- LCM of 6, 72, and 120: 360
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