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 that 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 determine the prime factorization of each number.

- **18** can be factored into \(2 \times 3^2\).
- **24** can be factored into \(2^3 \times 3\).
- **32** can be factored into \(2^5\).

Now, we take the highest power of each prime factor:
- For \(2\), the highest power is \(2^5\) (from 32).
- For \(3\), the highest power is \(3^2\) (from 18).

Thus, the LCM is:
\[ LCM = 2^5 \times 3^2 = 32 \times 9 = 288. \]

### Step 2: Identify the smallest 4-digit number
The smallest 4-digit number is 1000.

### Step 3: Find the smallest 4-digit number divisible by 288
To find the smallest 4-digit number that is divisible by 288, we divide 1000 by 288 and round up to the nearest whole number:
\[ \frac{1000}{288} \approx 3.4722. \]
Rounding up gives us 4.

### Step 4: Multiply the LCM by this whole number
Now, we multiply 288 by 4:
\[ 288 \times 4 = 1152. \]

### Step 5: Verify that 1152 is a 4-digit number
1152 is indeed a 4-digit number.

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

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