The smallest four-digit number which is exactly divisible by 18, 32 and 48 is:

Options

A. 1252
B. 1156
C. 1256
D. 1152

Answer

To find the smallest four-digit number that is exactly divisible by 18, 32, and 48, we can follow these steps:

### Step 1: Find the Least Common Multiple (LCM) of the numbers 18, 32, and 48.
To find the LCM, we first determine the prime factorization of each number:
- **18** = 2 × 3²
- **32** = 2⁵
- **48** = 2⁴ × 3

Next, we take the highest power of each prime number:
- 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 four-digit number.
The smallest four-digit number is **1000**.

### Step 3: Determine the smallest four-digit number that is a multiple of 288.
To find the smallest four-digit number that is divisible by 288, we divide 1000 by 288 and round up to the nearest whole number:
\[ \frac{1000}{288} \approx 3.472 \]
Rounding up gives us **4**.

### Step 4: Multiply 288 by 4 to find the smallest four-digit number.
Now, we multiply:
\[ 288 \times 4 = 1152 \]

### Conclusion:
The smallest four-digit number that is exactly divisible by 18, 32, and 48 is **1152**.

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