GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 20 Oct 2019, 07:01

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.

Close

Request Expert Reply

Confirm Cancel

If for any positive integer x, d[x] denotes its smallest odd

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Intern
Intern
avatar
Joined: 22 Sep 2010
Posts: 5
If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post Updated on: 12 Jul 2013, 02:13
2
20
00:00
A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

31% (02:11) correct 69% (01:59) wrong based on 372 sessions

HideShow timer Statistics

If for any positive integer x, d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

(1) D[x] - d[x] = 0
(2) D[3x] = 3

Originally posted by reg123456 on 01 Nov 2010, 08:58.
Last edited by Bunuel on 12 Jul 2013, 02:13, edited 1 time in total.
Edited the question and added the OA
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58434
Re: gmat club test DS - special operations  [#permalink]

Show Tags

New post 01 Nov 2010, 09:13
5
4
If for any positive integer x, d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

First of all note that the smallest positive odd divisor of any positive integer is 1. Thus \(d[x]=1\) for any x.

(1) D[x] - d[x] = 0 --> \(D[x] - 1 = 0\) --> \(D[x] = 1\) --> x can be 1, so odd or \(2^n\), (2, 4, 8, ...), so even. Not sufficient.

(2) D[3x] = 3 --> again x can be 1, so odd, as the largest odd divisor of \(3x=3\) is 3 or x can be \(2^n\) (2, 4, 8, ...), so even, as the largest odd divisor of 3*2=6 or 2*4=12 is 3. Not sufficient.

(1)+(2) From (1) and (2) we have that x can be either 1, so odd or 2^n, so even. Not sufficient.

Answer: E.
_________________
General Discussion
Director
Director
avatar
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 854
Re: gmat club test DS - special operations  [#permalink]

Show Tags

New post 05 May 2011, 22:49
1
x = 1 and 2 passes the conditions in both a and b.

thus E
Manager
Manager
avatar
Joined: 05 Nov 2012
Posts: 138
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 28 Dec 2013, 14:38
Bunuel wrote:
If for any positive integer x, d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

First of all note that the smallest positive odd divisor of any positive integer is 1. Thus \(d[x]=1\) for any x.

(1) D[x] - d[x] = 0 --> \(D[x] - 1 = 0\) --> \(D[x] = 1\) --> x can be 1, so odd or \(2^n\), (2, 4, 8, ...), so even. Not sufficient.

(2) D[3x] = 3 --> again x can be 1, so odd, as the largest odd divisor of \(3x=3\) is 3 or x can be \(2^n\) (2, 4, 8, ...), so even, as the largest odd divisor of 3*2=6 or 2*4=12 is 3. Not sufficient.

(1)+(2) From (1) and (2) we have that x can be either 1, so odd or 2^n, so even. Not sufficient.

Answer: E.

Hello Bunuel, question mentions that x is positive integer but did not mention d[x] to be smallest positive odd divisor right?! how can you consider d[x] to be 1? I want to consider d[x] as negative of highest positive odd divisor!
However, answer to the question will remain the same E though.
Intern
Intern
avatar
Joined: 03 Oct 2012
Posts: 8
Concentration: Strategy, Finance
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 12 Nov 2015, 10:47
1
nitinj025 wrote:
HI,
For the First Condition to Satisfy.
i)D[x]-d[x] =0 the value should be a prime number . Which, directly states that value is not even.
ii) D[3x] =1 the value can only be 1 for x. Which even states that the value is not even.
Please, Can you explain my doubt.
Regards,
NJ


Hi NJ,

1. D[x] - d[x] = 0 implies that the smallest odd divisor and largest odd divisor are the same. This divisor can also be 1. i.e. not even.
For x = 1, D[x] = 1 and d[x] = 1 => x is not even
For x = 2, D[x] = 1 and d[x] = 1 => x is even
Insufficient.

You have incorrectly read the statement 2 to be D[3x] = 1.
2. D[3x] = 3
For x = 1, D[3x] = 3, x is not even
For x = 2, D[3x] = 3 (because 3x = 6, factors of 6 = 1, 2, 3, 6 i.e. largest odd factor =3) i.e. x is even.
For x = 3, D[3x] = 9 (because 3x = 9, factors of 6 = 1, 3, 9 i.e. largest odd factor =9) - Not relevant as it does not satisfy D[3x] = 3.
Insufficient.

Together, based on statement 2, only possible values for x are 1 and 2.
x = 1, d[x] = 1, D[x] = 1, D[3x] = 3, x is odd
x = 2, d[x] = 1, D[x] = 1, D[3x] = 1, x is even
Therefore, together insufficient. i.e Answer is E.

Hope this helps.
Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 8017
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 15 Nov 2015, 11:01
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If for any positive integer x, d[x] denotes its smallest odd divisor and D[x] denotes its largest odd divisor, is x even?

(1) D[x] - d[x] = 0
(2) D[3x] = 3

There is one variable (x) and 2 more equations are given by the 2 conditions.
For condition 1, D[x]=d[x], the answer is 'yes' for x=1, but 'no' for x=2. This is insufficient.
For condition 2, D[3x]=3, this also gives 'yes' for x=1, but 'no' for x=2. This is insufficient.
Looking at them together, it still gives 'yes' for x=1, but 'no' for x=2. The answer is therefore (E).

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $79 for 1 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"
Intern
Intern
avatar
B
Joined: 06 Feb 2018
Posts: 16
Reviews Badge
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 26 Mar 2018, 11:32
Bunuel, how can x be 2^n?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58434
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 26 Mar 2018, 11:37
1
Non-Human User
User avatar
Joined: 09 Sep 2013
Posts: 13315
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

Show Tags

New post 13 Apr 2019, 09:36
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Club Bot
Re: If for any positive integer x, d[x] denotes its smallest odd   [#permalink] 13 Apr 2019, 09:36
Display posts from previous: Sort by

If for any positive integer x, d[x] denotes its smallest odd

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne