How many ordered pairs of $(m, n)$ integers satisfy $\frac{m}{12} = \frac{12}{n}$?

Consider the following statements:
$I$: The number of non-trivial even divisors of the number $N = 2^{\alpha_1} 3^{\alpha_2} 4^{\alpha_3} 5^{\alpha_4} 6^{\alpha_5}$ is $(\alpha_1+2\alpha_3+\alpha_5)(\alpha_2+\alpha_5+1)(\alpha_4+1)-1$.
$II$: The number of non-trivial odd divisors of the number $N = 2^{\alpha_1} 3^{\alpha_2} 4^{\alpha_3} 5^{\alpha_4} 6^{\alpha_5}$ is $\alpha_2+\alpha_4+\alpha_5+\alpha_2\alpha_4+\alpha_4\alpha_5$. Then:

The constant term in the expansion of $\left(2x + \frac{1}{x^7} + 3x^2\right)^5$ is $........$.

Number of ways in which the number $831600$ can be split into two factors which are relatively prime is

The number of $3$-digit numbers that are divisible by $2$ and $3$,but not divisible by $4$ and $9$,is

The largest non-negative integer $k$ such that $24^k$ divides $13!$ is