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?


Hint: Factor $a$ into primes; it's evident that these primes must be $2$, $3$, and $5$ (why?). Then $a = 2^d \cdot 3^b \cdot 5^c$, so that
$$a^4 = 2^{4d} \cdot 3^{4b} \cdot 5^{4c}$$
What conditions on $d, b, c$ are necessary and sufficient to make this a divisor of $2^{10} \cdot 3^8 \cdot 5^6$?
