What is the smallest square number which is divisible by 4, 6 and 32?

The correct option is 576.

To find the smallest square number that is divisible by 4, 6, and 32, we first need to find the Least Common Multiple (LCM) of these three numbers, and then find the smallest multiple of this LCM that is a perfect square.

Step 1: Find the prime factorization of each number
Let us write the prime factorization for each of the given numbers:
$$4=2^2$$
$$6=2\times 3$$
$$32=2^5$$

Step 2: Find the LCM of 4, 6, and 32
The LCM is calculated by taking the highest power of each prime factor that appears in the factorizations:
- The prime factors present are 2 and 3.
- The highest power of 2 is $2^5$ (from 32).
- The highest power of 3 is $3^1$ (from 6).
Multiplying these together gives the LCM:
$$\text{LCM}=2^5\times 3^1=32\times 3=96$$
Any number divisible by 4, 6, and 32 must be a multiple of 96.

Step 3: Make the LCM a perfect square
For a number to be a perfect square, all the exponents in its prime factorization must be even numbers. Let's look at the prime factorization of 96:
$$96=2^5\times 3^1$$
Here, the exponent of 2 is 5 (which is odd) and the exponent of 3 is 1 (which is also odd). To make both exponents even, we need to multiply 96 by one factor of 2 and one factor of 3:
$$2\times 3=6$$
Now, we multiply 96 by 6 to get the smallest perfect square:
$$96\times 6=576$$
We can verify that 576 is indeed a perfect square:
$$576={24}^2$$
Thus, the smallest square number divisible by 4, 6, and 32 is 576.