Question

Prove that the remainder when $1!+2!+.....+1000!$ is divided by $15$ be $3$.

Answer 1

We have $5!=\mathrm{5.4.3.2.1}=120$ is perfectly divisible by $15$.

Again $(5+n)!$[$n\in N$] has a factor $15$, this gives remainder $0$ when divided by $15$.

Now reminder will come only from $1!+2!+3!+4!=33$.

This leaves remainder $3$ when divided by $15$.