The least number which is a perfect square and is also divisible by 10, 12, 15 and 18 is

Options

A. 3600
B. 2500
C. 1600
D. 900

Answer

To find the least number which is a perfect square and is also divisible by 10, 12, 15, and 18, we can follow these steps:

### Step 1: Find the LCM of the given numbers
To find the least common multiple (LCM) of 10, 12, 15, and 18, we first need to factor each number into its prime factors.

- **10 = 2 × 5**
- **12 = 2² × 3**
- **15 = 3 × 5**
- **18 = 2 × 3²**

### Step 2: Identify the highest powers of each prime factor
Next, we take the highest power of each prime factor from the factorizations:

- The highest power of **2** is **2²** (from 12).
- The highest power of **3** is **3²** (from 18).
- The highest power of **5** is **5¹** (from 10 and 15).

### Step 3: Calculate the LCM
Now, we can calculate the LCM by multiplying these highest powers together:

\[ \text{LCM} = 2² × 3² × 5¹ = 4 × 9 × 5 \]

Calculating this step-by-step:

1. \(4 × 9 = 36\)
2. \(36 × 5 = 180\)

So, the LCM of 10, 12, 15, and 18 is **180**.

### Step 4: Find the smallest perfect square multiple of the LCM
Now, we need to find the smallest perfect square that is a multiple of 180. A perfect square must have even powers of all prime factors.

The prime factorization of 180 is:

\[ 180 = 2² × 3² × 5¹ \]

To make this a perfect square, we need to adjust the powers of the prime factors:

- 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 another **5** to make it even (2).

Thus, we need to multiply 180 by 5 to make it a perfect square:

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

### Conclusion
The least number which is a perfect square and is also divisible by 10, 12, 15, and 18 is **900**.