(Smallest common multiple of 18, 24, 48) `xx` (Highest common factor of 18, 24 and 48) is

Options

A. 846
B. 864
C. 380
D. 684

Answer

To solve the problem of finding the product of the smallest common multiple (LCM) of 18, 24, and 48 with the highest common factor (HCF) of the same numbers, we will follow these steps:

### Step 1: Find the LCM of 18, 24, and 48

To find the LCM, we can use the prime factorization method.

1. **Prime Factorization**:
- 18 = 2 × 3²
- 24 = 2³ × 3
- 48 = 2⁴ × 3

2. **Identify the highest power of each prime**:
- For the prime number 2, the highest power is 2⁴ (from 48).
- For the prime number 3, the highest power is 3² (from 18).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^4 \times 3^2 = 16 \times 9 = 144 \]

### Step 2: Find the HCF of 18, 24, and 48

To find the HCF, we again use the prime factorization.

1. **Prime Factorization** (as above):
- 18 = 2 × 3²
- 24 = 2³ × 3
- 48 = 2⁴ × 3

2. **Identify the lowest power of each common prime**:
- For the prime number 2, the lowest power is 2¹ (from 18).
- For the prime number 3, the lowest power is 3¹ (from 24 and 48).

3. **Calculate the HCF**:
\[ \text{HCF} = 2^1 \times 3^1 = 2 \times 3 = 6 \]

### Step 3: Multiply the LCM and HCF

Now, we multiply the LCM and HCF that we found:

\[ \text{Result} = \text{LCM} \times \text{HCF} = 144 \times 6 \]

Calculating this:

1. \( 144 \times 6 = 864 \)

### Final Answer:

The final answer is \( 864 \).

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