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August 16th, 2018, 04:14 AM  #1 
Newbie Joined: Aug 2018 From: TÃ¼rkiye Posts: 1 Thanks: 0  The number of a number in the prime factorization of a factorial
How do we find, for example, how many '5's there are in the prime factorization of n! ? I've read that it's floor(n/5), but why is that? Thanks a lot. Last edited by skipjack; August 16th, 2018 at 01:26 PM. 
August 16th, 2018, 05:12 AM  #2 
Member Joined: Aug 2015 From: Montenegro (Podgorica) Posts: 37 Thanks: 3 
Because 5 must appear at least once in every fifth number from 1 to n. Sent from my HUAWEI SCLL04 using Tapatalk 
August 16th, 2018, 07:49 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,505 Thanks: 1741 
So it would be at least floor(n/5), not necessarily exactly floor(n/5).

August 16th, 2018, 12:14 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,581 Thanks: 610 
The answer is $\sum_{k=1}^{\infty}\lfloor \frac{n}{5^k}\rfloor$. Note that the sum is finite since terms where $5^k \gt n$ are all 0.


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factorial, factorization, number, prime 
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