If $(24, 92) = 24m + 92n$,then $(m, n)$ is

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 greatest value of $x$ satisfying $21 \equiv 385 \pmod{x}$ and $587 \equiv 167 \pmod{x}$ is

If $1000! = 3^n \times m$ where $m$ is an integer not divisible by $3$, then $n = $

The number of $6$-digit numbers of the form $ababab$ (in base $10$) each of which is a product of exactly $6$ distinct primes is

If $(1-x^3)^{10} = \sum_{r=0}^{10} a_r x^r (1-x)^{30-2r}$, then $\frac{9a_9}{a_{10}}$ is equal to . . . . . . .