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.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this communityWhat 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.
If $n\ge 4$, then $4!=24$ divides $n!$ $-$ in particular $12$ divides $n!$ when $\ge 4$.
Thus $$ 1!+2!+\cdots+1000!=1!+2!+3! \!\!\!\!\pmod{12}=9\!\!\!\!\pmod{12}. $$
Hint: Every term from $12!$ onward is divisible by $12$, so they don't matter.