Can you explain why the floor equation is used for multiple primes (such as 2) several times, and how to write the induction proof for this question? Doe the proof require I use summation notation? If so why? does it account for the powers of 2? (Any clarity the writing an induction proof like this will be appreciated)

My basic understanding:

(using 19!)

For 19! the floor function 19/2 is 9, 19/4 <--- 2^2 = 4, 19/8 <----2^3 = 2 ....

why do I have to do each separately? Isn't there overlap?

Do I have to determine what happens with multiples of 3 also? and other values? Why?

Here is the question:

Letnbeanaturalnumber,andletpbeprime.Whatisthelargestpowerofpthatdividesn!(n factorial)?

(Youshouldgetstuckhereandstartplayingwithexamples!)

Onceyougureitout,thinkofthefollowingquestions:whatisthelargestpowerof10thatdivides(a) 50!,(b)1000!?