<s>[INST] For how many integers $a$ is $\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$ an integer? In Mathleague $11316$ Target #$4$, the question is:

For how many integers $a$ is $$\frac{2^{10} \cdot 3 ^8 \cdot 5^6}{a^4}$$ an integer?

 [/INST] Clearly the number
$$ q =\frac{2^{10} 3^8 5^6 }{a^4} = \left(  \frac{2^{5} 3^4 5^3 }{a^2}   \right)^2
$$
is a perfect square, and so
$$
\sqrt{q} = \frac{2^{5} 3^4 5^3 }{a^2}
$$ is rational. Now in order for this to be an integer, the denominator must divide the numerator. Therefore $a^2$ must divide $2^53^45^3$. Now the greatest perfect square dividing $2^53^45^3$ is $2^4 3^4 5^2$, and so $a^2$ can be any divisor of this number. Therefore $a$ can be any integer divisor of its square root, $2^2 3^2 5$. There are $3(3)(2)=18$ positive divisors of this number, and so there are $36$ signed possibilities for $a$. So the answer is 36.
</s>