Find the least possible perfect square number which is exactly divisible by 6, 40, 49 and 75.

Answer

To find the least possible perfect square number which is exactly divisible by 6, 40, 49, and 75, we can follow these steps:

### Step 1: Prime Factorization of Each Number
First, we will perform the prime factorization of each of the numbers:

- **6** can be factored as:
\( 6 = 2^1 \times 3^1 \)

- **40** can be factored as:
\( 40 = 2^3 \times 5^1 \)

- **49** can be factored as:
\( 49 = 7^2 \)

- **75** can be factored as:
\( 75 = 3^1 \times 5^2 \)

### Step 2: Finding the LCM
Next, we will find the Least Common Multiple (LCM) of these numbers by taking the highest power of each prime factor that appears in the factorizations:

- For \(2\): The highest power is \(2^3\) (from 40).
- For \(3\): The highest power is \(3^1\) (from both 6 and 75).
- For \(5\): The highest power is \(5^2\) (from 75).
- For \(7\): The highest power is \(7^2\) (from 49).

Thus, the LCM can be calculated as follows:
\[ \text{LCM} = 2^3 \times 3^1 \times 5^2 \times 7^2 \]

### Step 3: Calculating the LCM
Now, we will compute the LCM:
\[ \text{LCM} = 8 \times 3 \times 25 \times 49 \]

Calculating step-by-step:
- \(8 \times 3 = 24\)
- \(24 \times 25 = 600\)
- \(600 \times 49 = 29400\)

So, the LCM of 6, 40, 49, and 75 is \(29400\).

### Step 4: Ensuring the Result is a Perfect Square
Now, we need to ensure that this LCM is a perfect square. A number is a perfect square if all the exponents in its prime factorization are even.

The prime factorization of \(29400\) is:
\[ 29400 = 2^3 \times 3^1 \times 5^2 \times 7^2 \]

Here, the exponents are:
- \(2^3\) (odd)
- \(3^1\) (odd)
- \(5^2\) (even)
- \(7^2\) (even)

To make all the exponents even, we need to multiply by:
- \(2^{1}\) (to make \(2^3\) into \(2^4\))
- \(3^{1}\) (to make \(3^1\) into \(3^2\))

Thus, we need to multiply \(29400\) by \(2^1 \times 3^1 = 6\).

### Step 5: Calculate the Perfect Square
Now, we compute:
\[ \text{Perfect Square} = 29400 \times 6 = 176400 \]

### Conclusion
The least possible perfect square number which is exactly divisible by 6, 40, 49, and 75 is:
\[ \boxed{176400} \]