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

It is currently 18 Dec 2018, 09:22

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
Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
  • Happy Christmas 20% Sale! Math Revolution All-In-One Products!

     December 20, 2018

     December 20, 2018

     10:00 PM PST

     11:00 PM PST

    This is the most inexpensive and attractive price in the market. Get the course now!
  • Key Strategies to Master GMAT SC

     December 22, 2018

     December 22, 2018

     07:00 AM PST

     09:00 AM PST

    Attend this webinar to learn how to leverage Meaning and Logic to solve the most challenging Sentence Correction Questions.

If x ≠ 0, does x have an odd number of factors?

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

Hide Tags

Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 51280
If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 31 Jul 2018, 21:00
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

59% (00:55) correct 41% (01:12) wrong based on 135 sessions

HideShow timer Statistics

Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7112
If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 31 Jul 2018, 21:28
If x ≠ 0, does x have an odd number of factors?

ODD number of factors :- Only perfect square have odd factors
so the question is :- Is x a perfect square?

(1) √x is an integer.
let √x=y, where y is an integer
so \(x=y^2\), thus x is a perfect square
sufficient

(2) x^2 is an integer.
if x is an integer, ans can be YES
but if x is not an integer, for example \(x^2=3......x=\sqrt{3}\)... ans is NO
insuff

A
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Director
Director
User avatar
S
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 516
Location: India
Premium Member
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 31 Jul 2018, 23:27
2
Hi,

This question, if you know below properties of perfect square numbers then its just a twenty second problem.

1). A perfect square will always have an odd number of factors

2). A square of a prime number will always have exactly 3 factors

3). A perfect square will have an odd number of odd factors and even number of even factors.

Let’s say if one student could not recollect this during the exam,then its always good to try some numbers.

Let’s say for an example,

If x = 2,3 then it has even number of factors. Infact all the prime numbers as even number of factors.

If x is an odd number (not prime), let’s say x = 15, then x has 1,3,5,15 four factors.

If x is an even number (not prime), let’s say x = 20, then x has 1,2,4,5,10 and 20 six factors.

Now lets x is 4 or 9, then the factors are 1,2, 4 and for 9 we have 1,3,9.

We can see that, perfect square numbers have odd number of factors.

This is infact because,

When we prime factorize as perfect square number, always the power of primes is even.

Then the number of factors would be odd.

For example, 36 = 2^2 * 3^2

So, the number of factors is equal to (2+1) * (2+1) = 9 factors.

During the exam, it won’t take more than 40 or 50 secs to check the possibilities(But I would still recommend to know the properties as these are commonly asked questions in GMAT).

Statement I is sufficient:

√x is an integer.

So, x is a perfect square number.

So sufficient.

Statement II is insufficient:

x^2 is an integer.

First thing is “x” here may or may not be an integer.

Let’s say x = √2 is not an integer. But x^2 is not an integer.

Even if we consider “x” is an integer, we are not still sure whether “x” is a perfect square number or not.

So not sufficient.

So the answer is A here.
_________________

GMAT Mentors
Image

Intern
Intern
avatar
B
Joined: 30 Apr 2018
Posts: 19
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 03:51
chetan2u

I had a doubt in the part where you mention -
(2) x^2 is an integer.
if x is an integer, ans is YES
but if x is not an integer, or example x2=3......x=3√x2=3......x=3... ans is NO
insuff

if i assume = x = 16 i.e. integer and because 16 is a perfect sq. we can conclude = odd no. of integers
However if i assume = x = 2 i.e. integer but because 2 is not a perfect sq. we can conclude = no odd no. of integers,

hence unless we know what the exact value of x is (irrespective of integer or not) = would it not be impossible to state whether or not x having odd # of factors?

if i am missing out on a concept, please guide .
TIA
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7112
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 04:11
NidSha wrote:
chetan2u

I had a doubt in the part where you mention -
(2) x^2 is an integer.
if x is an integer, ans is YES
but if x is not an integer, or example x2=3......x=3√x2=3......x=3... ans is NO
insuff

if i assume = x = 16 i.e. integer and because 16 is a perfect sq. we can conclude = odd no. of integers
However if i assume = x = 2 i.e. integer but because 2 is not a perfect sq. we can conclude = no odd no. of integers,

hence unless we know what the exact value of x is (irrespective of integer or not) = would it not be impossible to state whether or not x having odd # of factors?

if i am missing out on a concept, please guide .
TIA



I think you mean x^2 =16 then x=4... X is an integer so x^2 has odd factors
And x^2 =2 then x=√2, here we do not take X as an integer, that is why it is not perfect square.

If you meant X is integer..
X=16, so x^2 =16^2, thus 16^2 is square
X=2 so x^2=2^2=4, again a perfect square
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Intern
Intern
avatar
B
Joined: 30 Apr 2018
Posts: 19
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 04:38
chetan2u

sorry, i couldn't express concrete,

Ask = does x have odd no. of factors?

i am assuming that i the concept here is right => perfect squares have odd no. of factors | non perfect squares we cant say unless we know the exact number

Hence,
1. if x^sq = 256 ; hence x = 16 and integer; and yes, going by the conceptual assumption because x = perfect square (as 4*4 = 16) => x has odd no. of factors
2. However, if x^sq = 4 ; hence x = 2 and integer; and again, going by the conceptual assumption because x = not a perfect square => x having odd no. of factors is not clear

by the above two example, shouldn't Statement (2) of the question be insufficient irrespective of x being an integer or not?

Thanks for your prompt responses chetan2u
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7112
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 04:55
NidSha wrote:
chetan2u

sorry, i couldn't express concrete,

Ask = does x have odd no. of factors?

i am assuming that i the concept here is right => perfect squares have odd no. of factors | non perfect squares we cant say unless we know the exact number

Hence,
1. if x^sq = 256 ; hence x = 16 and integer; and yes, going by the conceptual assumption because x = perfect square (as 4*4 = 16) => x has odd no. of factors
2. However, if x^sq = 4 ; hence x = 2 and integer; and again, going by the conceptual assumption because x = not a perfect square => x having odd no. of factors is not clear

by the above two example, shouldn't Statement (2) of the question be insufficient irrespective of x being an integer or not?

Thanks for your prompt responses chetan2u



Yes you are absolutely correct in your observation..
If x^2 is an integer, X may be a square or may not be .
However if it is not an integer, it is always NO.
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


GMAT online Tutor

Intern
Intern
avatar
B
Joined: 30 Apr 2018
Posts: 19
Re: If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 05:04
chetan2u

Thanks a lot for your help :)
GMATH Teacher
User avatar
G
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 552
If x ≠ 0, does x have an odd number of factors?  [#permalink]

Show Tags

New post 29 Aug 2018, 05:35
Bunuel wrote:
If x ≠ 0, does x have an odd number of factors?

(1) √x is an integer.

(2) x^2 is an integer.

Beautiful problem!

\(?\,\,\,\,:\,\,\,\,\,\# \,\,\left( {{\text{positive}}} \right)\,\,{\text{factors}}\,\,{\text{odd}}\,\,{\text{?}}\)

Important: it is not known (pre-statements) whether x is an integer (this is part of the problem)!
If x is not an integer, the answer (=focus) is <NO>, because non-integers don´t have factors (by definition, factors are related to integers only)!

Statement (2) will explore that. Let´s start with it!

\(\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\,Take\,\,x = \sqrt 2 \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\, \hfill \\
\,Take\,\,x = 1\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\
\end{gathered} \right.\)

\(\left( 1 \right)\,\,\sqrt x \,\,\,\operatorname{int} \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}
x \geqslant 0\,\,\,\,{\text{implicitly}}\,\,\,\,\,\,\mathop \Rightarrow \limits^{x\,\, \ne \,\,0} \,\,\,\,\,x > 0 \hfill \\
x\,\, = {\left( {\sqrt x } \right)^2}\,\, = \,\,{\operatorname{int} ^{\,2}} = {\text{perfect}}\,\,{\text{square}} \hfill \\
\end{gathered} \right.\,\)

\(x\,\,{\text{perfect}}\,\,{\text{square}}\,\,\, \geqslant \,\,\,{\text{1}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,\,x = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\, \hfill \\
\,\,x \geqslant 4\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{x}}\,{\text{ = }}\,{\text{prime}}{{\text{s}}^{\,{\text{even}}\,{\text{powers}}}}\,\,\, \hfill \\
\end{gathered} \right.\)

\({\text{x}}\,{\text{ = }}\,{\text{prime}}{{\text{s}}^{\,\boxed{{\text{even}}\,\,{\text{powers}}}}}\,\,\,\,\, \Rightarrow \,\,\,\,? = \left( {\boxed{{\text{even}}}\, + 1} \right)\,\, \cdot \left( {\boxed{{\text{even}}}\, + 1} \right) \cdot \ldots \cdot \left( {\boxed{{\text{even}}}\, + 1} \right) = {\text{odd}}\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\)

Important: the explanation above shows the reason why every nonzero perfect square has an odd number of positive factors.

The above follows the notations and rationale taught in the GMATH method.
_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)
Course release PROMO : finish our test drive till 30/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!

GMAT Club Bot
If x ≠ 0, does x have an odd number of factors? &nbs [#permalink] 29 Aug 2018, 05:35
Display posts from previous: Sort by

If x ≠ 0, does x have an odd number of factors?

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


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.