What is the highest power of $3$ in $100!$?

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

Let $d(n)$ denote the number of divisors of $n$ including $1$ and itself. Then,$d(225)$,$d(1125)$,and $d(640)$ are

The number of positive integers $n$ such that $n+3$ divides $n^3-3$ is

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