### Divisibility & Induction

Food for thought

We denote the product of the first $20$ natural numbers by $20!$ and call this $20$ factorial.
1. What is the highest power of $5$ which is a divisor of $20$ factorial? Just how many factors does $20!$ have altogether?
2. Show that the highest power of $p$ that divides $500!$, where $p$ is a prime number and $p^t<500 < p^{t+1}$, is $\lfloor 500/p\rfloor+\lfloor 500/p^2\rfloor+\dotsb+\lfloor 500/p^t\rfloor,$ where $\lfloor x\rfloor$ (the floor of $x$) means to round $x$ down to the nearest integer. (For example, $\lfloor 3\rfloor=3$, $\lfloor 4.7\rfloor=4$, $\lfloor -2.7\rfloor=-3$, and so on.)
3. How many factors does $n!$ have?