What is the remainder when $1! + 2! + 3! + \dots + 100!$ is divided by 5?

Explanation

1. 01

   From $5!$ on, every factorial contains the factor 5, so it leaves remainder 0.

2. 02

   $1 + 2 + 6 + 24 = 33$, and $33 \bmod 5 = 3$.