(The smallest common multiple of 30, 45 and 60 between 200 and 400) `div` (The largest common factor of 15, 24 and 45) is equal to

Options

A. 180
B. 60
C. 90
D. 120

Answer

To solve the given problem step-by-step, we will first find the smallest common multiple (LCM) of the numbers 30, 45, and 60 that lies between 200 and 400. Then, we will find the largest common factor (HCF) of the numbers 15, 24, and 45, and finally, we will divide the two results.

### Step 1: Find the LCM of 30, 45, and 60

1. **Prime Factorization**:
- 30 = 2 × 3 × 5
- 45 = 3² × 5
- 60 = 2² × 3 × 5

2. **Identify the highest powers of each prime**:
- For 2: highest power is 2² (from 60)
- For 3: highest power is 3² (from 45)
- For 5: highest power is 5¹ (from all)

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

4. **Find the smallest LCM between 200 and 400**:
- The LCM we found is 180, which is not between 200 and 400.
- The next multiple of 180 is \( 180 × 2 = 360 \).

### Step 2: Find the HCF of 15, 24, and 45

1. **Prime Factorization**:
- 15 = 3 × 5
- 24 = 2³ × 3
- 45 = 3² × 5

2. **Identify the lowest powers of common primes**:
- The only common prime factor is 3.
- The lowest power of 3 is 3¹.

3. **Calculate the HCF**:
\[ \text{HCF} = 3 \]

### Step 3: Divide the LCM by the HCF

1. **Perform the division**:
\[ \frac{\text{LCM}}{\text{HCF}} = \frac{360}{3} = 120 \]

### Final Answer:
The result of the division is **120**.

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