Difference of (smallest common multiple of 18, 20, 30) and (smallest common multiple of 36, 15, 21) is

Options

A. 1500
B. 1080
C. 900
D. 1000

Answer

To solve the problem of finding the difference between the smallest common multiple (LCM) of two sets of numbers, we will follow these steps:

### Step 1: Find the LCM of the first set of numbers (18, 20, 30)

1. **Prime Factorization**:
- 18 = 2 × 3²
- 20 = 2² × 5
- 30 = 2 × 3 × 5

2. **Identify the highest power of each prime**:
- For 2: highest power is 2² (from 20)
- For 3: highest power is 3² (from 18)
- For 5: highest power is 5¹ (from 20 and 30)

3. **Calculate the LCM**:
\[ \text{LCM} = 2² × 3² × 5¹ = 4 × 9 × 5 \]
\[ = 36 × 5 = 180 \]

### Step 2: Find the LCM of the second set of numbers (36, 15, 21)

1. **Prime Factorization**:
- 36 = 2² × 3²
- 15 = 3 × 5
- 21 = 3 × 7

2. **Identify the highest power of each prime**:
- For 2: highest power is 2² (from 36)
- For 3: highest power is 3² (from 36)
- For 5: highest power is 5¹ (from 15)
- For 7: highest power is 7¹ (from 21)

3. **Calculate the LCM**:
\[ \text{LCM} = 2² × 3² × 5¹ × 7¹ = 4 × 9 × 5 × 7 \]
\[ = 36 × 5 × 7 = 180 × 7 = 1260 \]

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

1. **Calculate the difference**:
\[ \text{Difference} = \text{LCM of (18, 20, 30)} - \text{LCM of (36, 15, 21)} = 180 - 1260 \]
\[ = -1080 \]

### Final Answer:
The difference is **-1080**.

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