The least perfect square number divisible by 3,4,5,6,8 is

Options

A. 900
B. 1200
C. 2500
D. 3600

Answer

To find the least perfect square number that is divisible by 3, 4, 5, 6, and 8, we will follow these steps:

### Step 1: Find the LCM of the numbers
We need to find the least common multiple (LCM) of the numbers 3, 4, 5, 6, and 8.

- **Prime factorization**:
- 3 = 3^1
- 4 = 2^2
- 5 = 5^1
- 6 = 2^1 * 3^1
- 8 = 2^3

- **Take the highest power of each prime**:
- For 2: the highest power is 2^3 (from 8)
- For 3: the highest power is 3^1 (from 3 or 6)
- For 5: the highest power is 5^1 (from 5)

- **Calculate the LCM**:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^1 = 8 \times 3 \times 5 \]
\[ = 24 \times 5 = 120 \]

### Step 2: Ensure the LCM is a perfect square
The LCM we found is 120. To find the least perfect square divisible by 120, we need to adjust the prime factorization of 120 to make all the exponents even.

- **Prime factorization of 120**:
\[ 120 = 2^3 \times 3^1 \times 5^1 \]

- **Adjust the exponents to be even**:
- For 2^3, we need one more 2 to make it 2^4.
- For 3^1, we need one more 3 to make it 3^2.
- For 5^1, we need one more 5 to make it 5^2.

- **New perfect square**:
\[ \text{Perfect square} = 2^4 \times 3^2 \times 5^2 \]

### Step 3: Calculate the perfect square
Now, we will calculate the value of the perfect square:
\[ = (2^4) \times (3^2) \times (5^2) = 16 \times 9 \times 25 \]
Calculating step by step:
- First, calculate \(16 \times 9 = 144\)
- Then, \(144 \times 25 = 3600\)

### Conclusion
The least perfect square number that is divisible by 3, 4, 5, 6, and 8 is **3600**.

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