How many factors of $$2^4 \times 3^5 \times 10^4$$ are perfect squares which are greater than 1?
Correct Answer: 44
$$2^4 \times 3^5 \times 10^4$$
=$$2^4 \times 3^5 \times 2^4*5^4$$
=$$2^8 \times 3^5 \times 5^4$$
For the factor to be a perfect square, the factor should be even power of the number.
In $$2^8$$, the factors which are perfect squares are $$2^0, 2^2, 2^4, 2^6, 2^8$$ = 5
Similarly, in $$3^5$$, the factors which are perfect squares are $$3^0, 3^2, 3^4$$ = 3
In $$5^4$$, the factors which are perfect squares are $$5^0, 5^2, 5^4$$ = 3
Number of perfect squares greater than 1 = 5*3*3-1
=44
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