What is the remainder left after dividing $1! + 2! + 3! + ... + 100!$ by $5$? I have this question as a homework. 

What is the remainder left after dividing $1! + 2! + 3! + \cdots + 100!$ by $5$?

I tried this: I noticed that every $n! \equiv 0 \pmod{5}$ for every $n\geq 5$.
For $n < 5$:
$$\begin{align*}
1! &\equiv 1 \pmod{5} \\
2! &\equiv 2 \pmod{5} \\
3! &\equiv 1 \pmod{5} \\
4! &\equiv 4 \pmod{5} \\
\end{align*}$$
So $1! + 2! + \cdots + 100! \equiv 8 \equiv 3 \pmod{5}$.
Therefore the remainder is $3$. 
Am I thinking properly?  
Thanks a lot.

For the sake of an answer:
YES. You are thinking properly. What you did is nice and correct. 
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