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# CAT 2019 Quant Question with Solution 53

Question:
How many factors of $2^{4} \times 3^{5} \times 10^{4}$ are perfect squares which are greater than 1 ?

${{2}^{4}}\times {{3}^{5}}\times {{10}^{4}}={{2}^{8}}\times {{3}^{5}}\times {{5}^{4}}$

For perfect squares, we have to take only even powers of the prime factors of the number.

The number of ways 2’s can be used is 5 i.e. ${{2}^{0}},\ {{2}^{2}},\ {{2}^{4}},\ {{2}^{6}},\ {{2}^{8}}\$

The number of ways 3’s can be used is 3 i.e. ${{3}^{0}},\ {{3}^{2}},\ {{3}^{4}}\$

The number of ways 5’s can be used is 3 i.e. ${{5}^{0}},\ {{5}^{2}},\ {{5}^{4}}\$

Therefore, the total number of factors which are perfect squares = $5\times 3\times 3=45$

But this also includes the number 1. Hence excluding 1, the required number is 45-1=44.

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