How many different prime factors does $20!$ have? (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
To find the prime factors of $20!$, I need to consider which prime numbers are less than or equal to $20$, since those are the only ones that could possibly divide $20!$. The prime numbers less than or equal to $20$ are $2, 3, 5, 7, 11, 13, 17$, and $19$. Therefore, $20!$ has exactly $8$ different prime factors. 