Difference of (Smallest common multiple of 4, 5 and 6) and (Smallest common multiple of 5, 6 and 9) is

Options

A. 30
B. 45
C. 48
D. 60

Answer

To solve the problem of finding the difference between the smallest common multiple (LCM) of the sets (4, 5, 6) and (5, 6, 9), we will follow these steps:

### Step 1: Find the LCM of 4, 5, and 6

1. **Prime Factorization**:
- 4 = 2²
- 5 = 5¹
- 6 = 2¹ × 3¹

2. **Identify the highest power of each prime**:
- For 2: the highest power is 2² (from 4)
- For 3: the highest power is 3¹ (from 6)
- For 5: the highest power is 5¹ (from 5)

3. **Calculate the LCM**:
\[ \text{LCM}(4, 5, 6) = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60 \]

### Step 2: Find the LCM of 5, 6, and 9

1. **Prime Factorization**:
- 5 = 5¹
- 6 = 2¹ × 3¹
- 9 = 3²

2. **Identify the highest power of each prime**:
- For 2: the highest power is 2¹ (from 6)
- For 3: the highest power is 3² (from 9)
- For 5: the highest power is 5¹ (from 5)

3. **Calculate the LCM**:
\[ \text{LCM}(5, 6, 9) = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90 \]

### Step 3: Find the difference between the two LCMs

1. **Calculate the difference**:
\[ \text{Difference} = \text{LCM}(5, 6, 9) - \text{LCM}(4, 5, 6) = 90 - 60 = 30 \]

### Final Answer
The difference of the smallest common multiple of (4, 5, 6) and (5, 6, 9) is **30**.

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