Find the HCF and the LCM of 300, 360, 600.

Answer

To find the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 300, 360, and 600, we can follow these steps:

### Step 1: Prime Factorization
First, we need to find the prime factorization of each number.

- **For 300:**
- 300 = 3 × 100
- 100 = 10 × 10
- 10 = 2 × 5
- Therefore, 300 = 3 × 2 × 5 × 5 = 3 × 2 × 5²

- **For 360:**
- 360 = 36 × 10
- 36 = 6 × 6
- 6 = 2 × 3
- Therefore, 360 = 2² × 3² × 5

- **For 600:**
- 600 = 6 × 100
- 6 = 2 × 3
- 100 = 10 × 10
- 10 = 2 × 5
- Therefore, 600 = 2³ × 3 × 5²

So, we have:
- 300 = 3 × 2 × 5²
- 360 = 2³ × 3² × 5
- 600 = 2³ × 3 × 5²

### Step 2: Finding HCF
The HCF is found by taking the lowest power of all common prime factors.

- **Common prime factors:**
- For 2: The lowest power is 2¹ (from 300).
- For 3: The lowest power is 3¹ (from 300 and 600).
- For 5: The lowest power is 5¹ (from 360).

Thus,
HCF = 2¹ × 3¹ × 5¹ = 2 × 3 × 5 = 30.

### Step 3: Finding LCM
The LCM is found by taking the highest power of all prime factors present in any of the numbers.

- **Prime factors:**
- For 2: The highest power is 2³ (from 360 and 600).
- For 3: The highest power is 3² (from 360).
- For 5: The highest power is 5² (from 300 and 600).

Thus,
LCM = 2³ × 3² × 5² = 8 × 9 × 25.

Calculating this step-by-step:
- 8 × 9 = 72
- 72 × 25 = 1800

### Final Answers
- HCF of 300, 360, and 600 is **60**.
- LCM of 300, 360, and 600 is **1800**.

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