What is the units digit of the sum $1! + 2! + 3! + 4! + 5! + \cdots + 1000!$?
The units digit of a sum depends only on the units digits of the addends. So, to find the units digit of the sum, I only need to look at the units digits of the factorials. I know that for any $n \geq 5$, the units digit of $n!$ is 0, since it is a multiple of 10. Therefore, the only factorials that contribute to the units digit of the sum are $1!, 2!, 3!, 4!$. I can easily compute these: $1! = 1$, $2! = 2$, $3! = 6$, $4! = 24$. So, the units digit of the sum is the same as the units digit of $1 + 2 + 6 + 4$, which is 3. 