Given that, $ 2^{4} \times 3^{5} \times 10^{4}$
We can write in prime factor form.
$2^{4} \times 3^{5} \times (5*2)^{4}$
$ \Rightarrow 2^{4} \times 3^{5} \times 5^{4} \times 2^{4} $
$ \Rightarrow 2^{8} \times 3^{5} \times 5^{4} \quad \longrightarrow (1)$
A perfect square is a number that can be expressed as the product of two equal integers.(or) we can say that when the power of a factor is even.
- For $ 2^{8} \Rightarrow 2^{0}, 2^{2}, 2^{4}, 2^{6}, 2^{8} $ are perfect square.
- For $3^{5} \Rightarrow 3^{0}, 3^{2}, 3^{4}$ are perfect square.
- For $5^{4} \Rightarrow 5^{0}, 5^{2} , 5^{4} $ are perfect square.
So, the total number of factors, which has even power $= 5 \times 3 \times 3 = 45 $
We need number of factors, which is greater than $1.$ So, $ 2^{0} \times 3^{0} \times 5^{0} =1$ will be subtracted from the total number of factors.
Therefore, the total number of factors, which is greater than $1=45-1=44$
Correct Answer $: 44$