(Smallest common multiple of 20, 24, 30) / (Greatest common factor of 8, 24, 40) is

Options

A. 15
B. 20
C. 25
D. 30

Answer

To solve the problem, we need to find the smallest common multiple (LCM) of the numbers 20, 24, and 30, and then find the greatest common factor (GCF) of the numbers 8, 24, and 40. Finally, we will divide the LCM by the GCF.

### Step-by-Step Solution:

**Step 1: Find the LCM of 20, 24, and 30.**

1. **Prime Factorization:**
- 20 = 2² × 5
- 24 = 2³ × 3
- 30 = 2 × 3 × 5

2. **Identify the highest power of each prime factor:**
- For 2: highest power is 2³ (from 24)
- For 3: highest power is 3¹ (from 24 and 30)
- For 5: highest power is 5¹ (from 20 and 30)

3. **Calculate the LCM:**
\[ \text{LCM} = 2³ × 3¹ × 5¹ = 8 × 3 × 5 = 120 \]

**Step 2: Find the GCF of 8, 24, and 40.**

1. **Prime Factorization:**
- 8 = 2³
- 24 = 2³ × 3
- 40 = 2³ × 5

2. **Identify the lowest power of each common prime factor:**
- For 2: lowest power is 2³ (common in all three numbers)

3. **Calculate the GCF:**
\[ \text{GCF} = 2³ = 8 \]

**Step 3: Divide the LCM by the GCF.**
\[ \frac{\text{LCM}}{\text{GCF}} = \frac{120}{8} = 15 \]

### Final Answer:
The result of the expression \((\text{Smallest common multiple of } 20, 24, 30) / (\text{Greatest common factor of } 8, 24, 40)\) is **15**.

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