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?