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