Let $N = 3^{5} \times 5^{4} \times 6^{9} \times 7^{7} .$ The number of factors of $N$ which are perfect squares is

168

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180

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210

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480

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Solution

The correct option is C $210$
$N = 3^{5} \times 5^{4} \times 6^{9} \times 7^{7}$
$= 2^{9} \times 3^{14} \times 5^{4} \times 7^{7}$

A perfect square of $N$ is of the form $2^{a} \times 3^{b} \times 5^{c} \times 7^{d}$
where $a \in {0, 2, 4, 6, 8}$
$b \in {0, 2, 4, \dots, 14}$
$c \in {0, 2, 4}$
$d \in {0, 2, 4, 6}$

Total number is $5 \times 8 \times 3 \times 4 = 480$

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