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

 It is currently 23 Feb 2019, 00:55

### 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

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

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT RC Webinar

February 23, 2019

February 23, 2019

07:00 AM PST

09:00 AM PST

Learn reading strategies that can help even non-voracious reader to master GMAT RC. Saturday, February 23rd at 7 AM PT
• ### FREE Quant Workshop by e-GMAT!

February 24, 2019

February 24, 2019

07:00 AM PST

09:00 AM PST

Get personalized insights on how to achieve your Target Quant Score.

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

Author Message
TAGS:

### Hide Tags

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

### Show Tags

Updated on: 12 Jul 2013, 01:13
2
13
00:00

Difficulty:

95% (hard)

Question Stats:

32% (02:13) correct 68% (01:59) wrong based on 474 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, 07:58.
Last edited by Bunuel on 12 Jul 2013, 01:13, edited 1 time in total.
Edited the question and added the OA
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: gmat club test DS - special operations  [#permalink]

### Show Tags

01 Nov 2010, 08:13
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.

_________________
##### General Discussion
Director
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 997
Re: gmat club test DS - special operations  [#permalink]

### Show Tags

05 May 2011, 21:49
1
x = 1 and 2 passes the conditions in both a and b.

thus E
Manager
Joined: 05 Nov 2012
Posts: 146
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

### Show Tags

28 Dec 2013, 13:38
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.

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
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

12 Nov 2015, 09:47
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
Joined: 16 Aug 2015
Posts: 6985
GMAT 1: 760 Q51 V42
GPA: 3.82
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

### Show Tags

15 Nov 2015, 10:01
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 \$149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Intern
Joined: 06 Feb 2018
Posts: 16
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

### Show Tags

26 Mar 2018, 10:32
Bunuel, how can x be 2^n?
Math Expert
Joined: 02 Sep 2009
Posts: 53066
Re: If for any positive integer x, d[x] denotes its smallest odd  [#permalink]

### Show Tags

26 Mar 2018, 10:37
1
mahrah wrote:
Bunuel, how can x be 2^n?

D[x] denotes the largest odd divisor of x. (1) says that D[x] = 1, so the the largest odd divisor of x. This means that x is some power of 2: 2^0 = 1, 2^1 = 2, 2^2 = 4, ... The largest, and the only, odd divisor of all of those numbers is 1.
_________________
Re: If for any positive integer x, d[x] denotes its smallest odd   [#permalink] 26 Mar 2018, 10:37
Display posts from previous: Sort by