TITLE: What is the remainder when $1! + 2! + 3! +\cdots+ 1000!$ is divided by $12$?
QUESTION [14 upvotes]: What is the remainder when $$1! + 2! + 3! +\cdots+ 1000!$$ is divided by $12$.
I have tried to find the answer using the Binomial Theorem but that doesn't help.
How will we do this?
Please help.

REPLY [46 votes]: If $n\ge 4$, then $4!=24$ divides $n!$ $-$ in particular $12$ divides $n!$ when $\ge 4$. 
Thus
$$
1!+2!+\cdots+1000!=1!+2!+3! \!\!\!\!\pmod{12}=9\!\!\!\!\pmod{12}.
$$

REPLY [35 votes]: Hint: Every term from $12!$ onward is divisible by $12$, so they don't matter.