Find the smallest `4`-digit number which is divisible by `18, 24` and `32`

Options

A. 1052
B. 1152
C. 1512
D. 1125

Explanation

To find the smallest 4-digit number which is divisible by 18, 24, and 32, we will follow these steps:

### Step 1: Find the LCM of 18, 24, and 32
To find the least common multiple (LCM), we will first factor each number into its prime factors:

- **18** = 2 × 3²
- **24** = 2³ × 3
- **32** = 2⁵

Next, we take the highest power of each prime factor:

- For **2**, the highest power is 2⁵ (from 32).
- For **3**, the highest power is 3² (from 18).

Now, we can calculate the LCM:

\[ \text{LCM} = 2^5 \times 3^2 = 32 \times 9 = 288 \]

### Step 2: Find the smallest 4-digit number divisible by 288
The smallest 4-digit number is 1000. We need to find the smallest multiple of 288 that is greater than or equal to 1000.

To do this, we divide 1000 by 288:

\[ 1000 \div 288 \approx 3.472 \]

Now, we round this up to the next whole number, which is 4.

### Step 3: Multiply 288 by 4
Now, we multiply 288 by 4 to find the smallest 4-digit number:

\[ 288 \times 4 = 1152 \]

### Conclusion
Thus, the smallest 4-digit number which is divisible by 18, 24, and 32 is **1152**.

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