How to find this expression $1000! \mod 3^{300}$

Question

How to find this expression $(1000!\mod 3^{300})$?

Answer 1

$3$ goes into $1000!$ at least $300$ times, since it divides $3, 6, 9, \dots, 900$, and hence $3^{300} \mid 1000!$.

Answer 2

Using this, the highest power of $3$ in $1000!>300$

So, the remainder $=0$

Answer 3

Hint $\rm\ B^\color{#C00}A \mid (AB)!\, =\, 1\cdot\cdot\: B\,\cdot\cdot\: 2B\,\cdot\cdot\ 3B\,\cdots \color{#C00}AB$
thus $\:3^{300}\mid 900!\mid 1000!$