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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?