A number when multiplied by 5 becomes divisible by 12,18,24,32 & 40 is.

Options

A. 7200
B. 288
C. 480
D. 460

Answer

To solve the problem step by step, we need to find a number \( x \) such that when multiplied by 5, it becomes divisible by the numbers 12, 18, 24, 32, and 40.

### Step 1: Find the LCM of the given numbers
To determine the number \( x \), we first need to find the least common multiple (LCM) of the numbers 12, 18, 24, 32, and 40.

1. **Prime Factorization**:
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
- \( 24 = 2^3 \times 3^1 \)
- \( 32 = 2^5 \)
- \( 40 = 2^3 \times 5^1 \)

2. **Determine the highest powers of each prime**:
- For \( 2 \): The highest power is \( 2^5 \) (from 32).
- For \( 3 \): The highest power is \( 3^2 \) (from 18).
- For \( 5 \): The highest power is \( 5^1 \) (from 40).

3. **Calculate the LCM**:
\[ \text{LCM} = 2^5 \times 3^2 \times 5^1 = 32 \times 9 \times 5 \]
\[ = 2880 \]

### Step 2: Set up the equation
We know that \( 5x \) must be equal to the LCM, which is 2880. Thus, we can set up the equation:
\[ 5x = 2880 \]

### Step 3: Solve for \( x \)
To find \( x \), we divide both sides of the equation by 5:
\[ x = \frac{2880}{5} \]
\[ x = 576 \]

### Conclusion
The number \( x \) is 576. Therefore, the number that when multiplied by 5 becomes divisible by 12, 18, 24, 32, and 40 is **576**.

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