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June 22nd, 2019, 09:06 AM   #1
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Divisible by 4

How many times can $\displaystyle 99!$ be divided by $\displaystyle 4$ ?
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June 22nd, 2019, 09:26 AM   #2
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As many times as you want, and the answer is the same every time.
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June 22nd, 2019, 09:48 AM   #3
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The highest integer power of 4 that divides 99! is the 47th.
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Last edited by skipjack; June 23rd, 2019 at 04:31 AM.
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June 22nd, 2019, 11:28 AM   #4
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I say it's $\displaystyle 4^{34}$
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June 22nd, 2019, 12:29 PM   #5
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Quote:
Originally Posted by shunya View Post
I say it's $\displaystyle 4^{34}$
Skipjack has the correct answer in post #3
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June 22nd, 2019, 12:56 PM   #6
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June 23rd, 2019, 12:39 AM   #7
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Maybe it must be posted in number theory forum.
The post#3 seems to be correct .
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June 23rd, 2019, 01:14 AM   #8
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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.
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June 23rd, 2019, 04:27 AM   #9
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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.
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June 23rd, 2019, 07:22 AM   #10
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Quote:
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How many times can $\displaystyle 99!$ be divided by $\displaystyle 4$ ?
$\frac{99!}{4}$ times
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