It is currently 21 Nov 2018, 05:38
My Tests

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

If 13!/2^x is an integer, which of the following represe

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Moderator
Moderator
User avatar
Joined: 18 Apr 2015
Posts: 4928
Followers: 74

Kudos [?]: 982 [0], given: 4518

CAT Tests
If 13!/2^x is an integer, which of the following represe [#permalink] New post 23 May 2017, 08:54
Expert's post
00:00

Question Stats:

48% (01:30) correct 51% (01:01) wrong based on 27 sessions


If \(\frac{13!}{2^x}\) is an integer, which of the following represents all possible values of x?


A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10
[Reveal] Spoiler: OA

_________________

Get the 2 FREE GREPrepclub Tests

Director
Director
Joined: 03 Sep 2017
Posts: 521
Followers: 1

Kudos [?]: 327 [0], given: 66

Re: If 13!/2^x is an integer, which of the following represe [#permalink] New post 22 Sep 2017, 05:48
We should start by the left constraint. 2^0 is 1 and every number is divisible by 1, thus the lower bound is 0, included. Thus, we already remain with two options A and C. Then, we have to check if 10 is included or not. To do so, it is easy to remember that 2^10 = 1024. Thus, we have to check if 13! is divisible by 1024 or 2^10. My idea is to check if there is a way to compute 1024 using the prime factors of 13!.

13! = 1*2*3*4*5*6*7*8*9*10*11*12*13 Then, let's rewrite every number is its prime factors, i.e 12! = 1*2*3*(2*2)*5*(2*3)*7*(2*2*2)*(3*3)*(2*5)*11*(2*2*3)*13. Then it's easy to notice than this long multiplication can be rewritten as 12!=1*2^10*3^5*5^2*7*11*13.

Thus if we divide 13! by 2^10 we see that 2^10 compares in the number above thus it simplifies and what remains is an integer. Thus, 10 is the upper bound of our interval and answer is A!
Director
Director
Joined: 20 Apr 2016
Posts: 743
Followers: 6

Kudos [?]: 500 [0], given: 86

CAT Tests
Re: If 13!/2^x is an integer, which of the following represe [#permalink] New post 22 Sep 2017, 08:01
Carcass wrote:


If \(\frac{13!}{2^x}\) is an integer, which of the following represents all possible values of x?


A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10


We can write 13! = 13*12*11*10*9*8*7*6*5*4*3*2*1

or 13! = \(13 * (3*2^2) * 11 * (2*5) * 9 * (2^3) * 7 * (2*3) * 5 * (2^2) * 3 * (2) * 1\)

So maximum power of 2 = 2^10 ( adding all powers of 2 we get 2^10, so the value of x has to be ≤ 10 to make the fraction as integer)

since \(2^0\) =1 and it is divisible by any 13! so

we can consider 0 ≤ x ≤ 10
_________________

If you found this post useful, please let me know by pressing the Kudos Button

1 KUDOS received
Target Test Prep Representative
User avatar
Status: Head GRE Instructor
Affiliations: Target Test Prep
Joined: 09 May 2016
Posts: 161
Location: United States
Followers: 4

Kudos [?]: 114 [1] , given: 0

Re: If 13!/2^x is an integer, which of the following represe [#permalink] New post 19 Dec 2017, 06:23
1
This post received
KUDOS
Expert's post
Carcass wrote:


If \(\frac{13!}{2^x}\) is an integer, which of the following represents all possible values of x?


A) 0 ≤ x ≤ 10

B) 0 < x < 9

C) 0 ≤ x < 10

D) 1 ≤ x ≤ 10

E) 1 < x < 10


Let’s determine the maximum number of factors of 2 within 13!. It would be very time consuming to list out each multiple of 2 in 13!. Instead, we can use the following shortcut in which we divide 13 by 2, and then divide the quotient of 13/2 by 2 and continue this process until we can no longer get a nonzero integer as the quotient.

13/2 = 6 (we can ignore the remainder)

6/2 = 3

3/2 = 1 (we can ignore the remainder)

Since 1/2 does not produce a nonzero quotient, we can stop.

The next step is to add our quotients; that sum represents the number of factors of 2 within 13!.

Thus, there are 6 + 3 + 1 = 10 factors of 2 within 13!.

So, x can be between zero and 10 inclusive.

Answer: A
_________________

Jeffery Miller
Head of GRE Instruction

GRE Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Re: If 13!/2^x is an integer, which of the following represe   [#permalink] 19 Dec 2017, 06:23
Display posts from previous: Sort by

If 13!/2^x is an integer, which of the following represe

  Question banks Downloads My Bookmarks Reviews Important topics  


GRE Prep Club Forum Home| About| Terms and Conditions and Privacy Policy| GRE Prep Club Rules| Contact

Powered by phpBB © phpBB Group

Kindly note that the GRE® test is a registered trademark of the Educational Testing Service®, and this site has neither been reviewed nor endorsed by ETS®.