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 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. Thanks from Maschke
 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. Thanks from idontknow 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
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Quote:
 Originally Posted by shunya I say it's $\displaystyle 4^{34}$
Skipjack has the correct answer in post #3

 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 Thanks from romsek and 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. Thanks from topsquark
 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. Thanks from Maschke
June 23rd, 2019, 07:22 AM   #10
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Quote:
 Originally Posted by idontknow How many times can $\displaystyle 99!$ be divided by $\displaystyle 4$ ?
$\frac{99!}{4}$ times

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