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.
divisibilityelementary-number-theoryfactorialintegers
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.
Best Answer
If $n\ge 4$, then $4!=24$ divides $n!$ $-$ in particular $12$ divides $n!$ when $\ge 4$.
Thus $$ 1!+2!+\cdots+1000!=1!+2!+3! \!\!\!\!\pmod{12}=9\!\!\!\!\pmod{12}. $$