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Factorial Fun

Age 16 to 18
Challenge Level

We denote the product of the first 20 natural numbers by 20! and call this 20 factorial.

(a) What is the highest power of 5 which is a divisor of 20 factorial? Just how many factors does 20! have altogether?

(b) 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 down to the nearest integer.  (For example, $\lfloor 3\rfloor=3$, $\lfloor 4.7\rfloor=4$, $\lfloor -2.7\rfloor=-3$, and so on.)

(c) How many factors does $n!$ have?