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How many factors $2^{4}\times3^{5}\times10^{4}$ are perfect squares which are greater than $1$ _______

Given that, $2^{4} \times 3^{5} \times 10^{4}$

We can write in prime factor from.

$2^{4} \times 3^{5} \times (5*2)^{4}$

$\Rightarrow 2^{4} \times 3^{5} \times 5^{4} \times 2^{4}$

$\Rightarrow 2^{8} \times 3^{5} \times 5^{4} \quad \longrightarrow (1)$

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 $: 44$

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