June 22nd, 2019, 09:06 AM  #1 
Senior Member Joined: Dec 2015 From: somewhere Posts: 729 Thanks: 98  Divisible by 4
How many times can $\displaystyle 99!$ be divided by $\displaystyle 4$ ?

June 22nd, 2019, 09:26 AM  #2 
Senior Member Joined: Oct 2009 Posts: 884 Thanks: 340 
As many times as you want, and the answer is the same every time.

June 22nd, 2019, 09:48 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
The highest integer power of 4 that divides 99! is the 47th.
Last edited by skipjack; June 23rd, 2019 at 04:31 AM. 
June 22nd, 2019, 11:28 AM  #4 
Senior Member Joined: Oct 2013 From: Far far away Posts: 431 Thanks: 18 
I say it's $\displaystyle 4^{34}$

June 22nd, 2019, 12:29 PM  #5 
Senior Member Joined: Sep 2015 From: USA Posts: 2,585 Thanks: 1430  
June 22nd, 2019, 12:56 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,839 Thanks: 653 Math Focus: Yet to find out. 
idontknow

June 23rd, 2019, 12:39 AM  #7 
Senior Member Joined: Dec 2015 From: somewhere Posts: 729 Thanks: 98 
Maybe it must be posted in number theory forum. The post#3 seems to be correct . 
June 23rd, 2019, 01:14 AM  #8 
Senior Member Joined: Aug 2012 Posts: 2,410 Thanks: 753 
There are 3 multiples of 32 = $2^5$: 32, 64, and 96. That accounts for 15 powers of 2. There are 6 multiples of 16, but we already counted 3, leaving 3 multiples of 16, which gives 12 powers of 2. There are 12 multiples of 8, but we already counted 3 + 3 = 6, leaving 6 multiples of 8, which gives 18 powers of 2. There are 24 multiples of 4, but we already counted 3 + 3 + 6 = 12, leaving 12 multiples of 4, which gives 24 powers of 2. There are 49 multiples of 2, but we already counted 3 + 3 + 6 + 12 = 24, leaving 25 multiples of 2. So there are 15 + 12 + 18 + 24 + 25 = 94 powers of 2, or 47 powers of 4. 
June 23rd, 2019, 04:27 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 21,028 Thanks: 2259 
As the multiples of 32 account for 16 powers of 2 (because 64 accounts for 6), there are 95 powers of 2 in total. The number of powers of 4 is still 47.

June 23rd, 2019, 07:22 AM  #10 
Senior Member Joined: Sep 2016 From: USA Posts: 670 Thanks: 440 Math Focus: Dynamical systems, analytic function theory, numerics  

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