Find the smallest perfect square number divisible by 12, 15 and 18

Options

A. A)900
B. B)1600
C. C)400
D. D)100

Answer

To find the smallest perfect square number that is divisible by 12, 15, and 18, we can follow these steps:

### Step 1: Find the LCM of the numbers
First, we need to find the least common multiple (LCM) of the three numbers: 12, 15, and 18.

- **Prime factorization of each number:**
- 12 = 2^2 × 3^1
- 15 = 3^1 × 5^1
- 18 = 2^1 × 3^2

- **Take the highest power of each prime:**
- For 2: highest power is 2^2 (from 12)
- For 3: highest power is 3^2 (from 18)
- For 5: highest power is 5^1 (from 15)

- **Calculate the LCM:**
\[ \text{LCM} = 2^2 × 3^2 × 5^1 = 4 × 9 × 5 = 180 \]

### Step 2: Factor the LCM to make it a perfect square
Now we need to ensure that the LCM (180) is a perfect square. A perfect square has all prime factors raised to an even power.

- **Prime factorization of 180:**
\[ 180 = 2^2 × 3^2 × 5^1 \]

- **Adjust the factors to make them even:**
- The power of 2 is already even (2).
- The power of 3 is already even (2).
- The power of 5 is odd (1), so we need to multiply by 5 to make it even.

### Step 3: Calculate the smallest perfect square
To make 180 a perfect square, we multiply it by 5:

\[ \text{Smallest perfect square} = 180 × 5 = 900 \]

### Conclusion
The smallest perfect square number that is divisible by 12, 15, and 18 is **900**.

---