1. Find the largest number $n$ such that (2004!)! is divisible by $((n!)!)!$.
Solution:
6

For positive integers $a, b$, we have
$$
a!\mid b!\Leftrightarrow a!\leq b!\Leftrightarrow a \leq b .
$$

Thus,
$$
((n!)!)!\mid(2004!)!\Leftrightarrow(n!)!\leq 2004!\Leftrightarrow n!\leq 2004 \Leftrightarrow n \leq 6 .
$$