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# If n is the greatest positive integer for which 2n is a fact

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If n is the greatest positive integer for which 2n is a fact [#permalink]

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25 Dec 2012, 10:05
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If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!
[Reveal] Spoiler: OA
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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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25 Dec 2012, 10:18
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In the questions where you are supposed to find out the maximum power of a prime factor which is a factor of n!, keep one only on thing in mind:
Let the prime number you are looking for be x, then:
$$n/x + n/x^2 + n/x^3.........n/x^k$$ where $$x^k$$ <=n.
If you apply this rule here,
$$10/2 +10/2^2 +10/2^3$$ or 5+2+1=8.
Hence maximum power of 2 in $$10!$$ will be 8.
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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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26 Dec 2012, 03:49
curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!

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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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09 Jan 2014, 10:44
We can count the total number of powers of 2 in 10!
There are 8 of them.So answer D.

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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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09 Jan 2014, 13:22
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If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Solution:
On a closer look this Qs is asking how many 2s are there in multiplication of 10!
Now, as we know 10! can be written as -- 10.9.8.7.6.5.4.3.2.1

As this is a small multiplication ( To review , not to do), we can check how many 2s we have in this..,
So, 10-1 (no.of 2s), 8-3(no.of 2s), 6-1(no.of 2s),4-2(no.of 2s),2-1(no.of 2s)

Hence, ans. D.

Hope this helps
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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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14 May 2015, 15:05
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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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14 May 2015, 23:06
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Expert's post
Hi All,

Many Test Takers get these types of questions wrong because they move too quickly through the work and don't do enough work on their pads. It's a relatively straight-forward prompt though...

We're essentially asked to find all the "2"s inside 10!

The 'key' to this question is to realize that some values have MORE THAN ONE 2 in them....

10! = (10)(9)(8)(7)(6)(5)(4)(3)(2)(1)

10 = 5x2 --> one 2
8 = 2x2x2 --> three 2s
6 = 3x2 --> one 2
4 = 2x2 --> two 2s
2 = 1x2 --> one 2

1+3+1+2+1 = eight 2s

[Reveal] Spoiler:
D

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Re: If n is the greatest positive integer for which 2n is a fact [#permalink]

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24 Mar 2016, 10:23
curtis0063 wrote:
If n is the greatest positive integer for which 2^n is a factor of 10!, then n =?

A. 2
B. 4
C. 6
D. 8
E. 10

Is any one can provide a solution for this question?It's from GWD. Thanks!

$$\frac{10}{2}$$ = 5
$$\frac{5}{2}$$ = 2
$$\frac{2}{2}$$ = 1

Now, 5+2+1 = 8

Hence the highest power of 2 that will divide 10! is 8
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Re: If n is the greatest positive integer for which 2n is a fact   [#permalink] 24 Mar 2016, 10:23
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