Check GMAT Club Decision Tracker for the Latest School Decision Releases https://gmatclub.com/AppTrack

 It is currently 23 May 2017, 23:43

### 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 June
Open Detailed Calendar

# Which of the following must be true if the square root of X

Author Message
TAGS:

### Hide Tags

Manager
Joined: 02 Nov 2012
Posts: 95
Location: India
Concentration: Entrepreneurship, Strategy
WE: Other (Computer Software)
Followers: 0

Kudos [?]: 45 [1] , given: 35

Which of the following must be true if the square root of X [#permalink]

### Show Tags

21 Dec 2012, 03:44
1
KUDOS
12
This post was
BOOKMARKED
00:00

Difficulty:

35% (medium)

Question Stats:

59% (01:57) correct 41% (00:51) wrong based on 368 sessions

### HideShow timer Statistics

Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

[Reveal] Spoiler:

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!
[Reveal] Spoiler: OA

_________________

TH

Give me +1 Kudos if my post helped!

Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [6] , given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

21 Dec 2012, 04:06
6
KUDOS
Expert's post
12
This post was
BOOKMARKED
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!

Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Hope it helps.
_________________
Moderator
Joined: 02 Jul 2012
Posts: 1223
Location: India
Concentration: Strategy
GMAT 1: 740 Q49 V42
GPA: 3.8
WE: Engineering (Energy and Utilities)
Followers: 124

Kudos [?]: 1483 [0], given: 116

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

21 Dec 2012, 04:17
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

[Reveal] Spoiler:

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!

I cannot be true since number of distinct factors of a square number is always odd. So we need to check only III. If III is true answer is D else answer can only be B.

Sum of distinct factors of a perfect square is always odd. Hence answer is D.

To answer your question, I believe the GMAT does not consider negative factors when it talks about factors of a number.
_________________

Did you find this post helpful?... Please let me know through the Kudos button.

Thanks To The Almighty - My GMAT Debrief

GMAT Reading Comprehension: 7 Most Common Passage Types

Intern
Joined: 16 Apr 2009
Posts: 16
Followers: 0

Kudos [?]: 14 [0], given: 5

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

22 Dec 2012, 01:28
Bunuel wrote:
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!

Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Hope it helps.

Thanks Bunuel, Could you please clarify the term "Distinct Factors"?
Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [0], given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

22 Dec 2012, 05:02
Drik wrote:
Thanks Bunuel, Could you please clarify the term "Distinct Factors"?

Not sure what to clarify: distinct=different, so for example distinct factors of 8 are 1, 2, and 8.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [0], given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

19 Jun 2013, 04:52
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All Must or Could be True Questions to practice: search.php?search_id=tag&tag_id=193

_________________
Manager
Status: Training
Joined: 03 Jun 2013
Posts: 90
GPA: 3.7
Followers: 3

Kudos [?]: 152 [0], given: 3

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

23 Jun 2013, 22:41
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.

Hey Bunuel or others:

Could you please show a proof for this?

Thanks
_________________

KUDOS please if my post was useful!

Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [0], given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

24 Jun 2013, 00:10
mattce wrote:
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.

Hey Bunuel or others:

Could you please show a proof for this?

Thanks

Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.
_________________
Manager
Status: Training
Joined: 03 Jun 2013
Posts: 90
GPA: 3.7
Followers: 3

Kudos [?]: 152 [0], given: 3

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

24 Jun 2013, 00:15
Bunuel wrote:
mattce wrote:
Bunuel wrote:
2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square.

Hey Bunuel or others:

Could you please show a proof for this?

Thanks

Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.

Haha, yeah I know that it's true by doing examples -- I was hoping for a formal proof though, if possible?
_________________

KUDOS please if my post was useful!

Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [0], given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

24 Jun 2013, 00:19
mattce wrote:
Bunuel wrote:
mattce wrote:

Hey Bunuel or others:

Could you please show a proof for this?

Thanks

Check for some perfect squares:
1 --> the sum factors = 1;
4 --> the sum factors = 7;
9 --> the sum factors = 13;
...

To see that the reverse is not always true check for 2 --> the sum factors = 3.

Hope it helps.

Haha, yeah I know that it's true by doing examples -- I was hoping for a formal proof though, if possible?

You can do it yourself using the formula for the sum of the factors given here: math-number-theory-88376.html
_________________
Intern
Joined: 24 Aug 2013
Posts: 1
Followers: 0

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

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

11 Oct 2013, 13:12
Bunuel wrote:
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!

Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Hope it helps.

Hi Bunuel,

My doubt was regarding the distinct factors.

For example

we take 16 -

Wont we consider the negative factors also?

Like for eg - for 16 they would be -1, 1, -2, 2, -4, 4, -8, 8, -16, 16 so that gives us an even number of distinct factors right?

Why wont we consider the negative in this case. The integers/factors with the negative sign are distinct too.

Will be grateful if you could clarify a little.

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 38846
Followers: 7721

Kudos [?]: 105951 [0], given: 11602

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

12 Oct 2013, 09:36
abhishekgulshan wrote:
Bunuel wrote:
th03 wrote:
Which of the following must be true if the square root of X is a positive integer?

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III

I have a doubt with the answer for this question. I believe that the right answer should be A. If X=4, then its factors are 1,2,4,-1,-2,-4. Should't we consider negative factors too??? Please explain. Thanks!

Factor is a "positive divisor" (at least on the GMAT). So, the factors of 4 are 1, 2, and 4 ONLY.

1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

According to this, only II and III must be true.

Hope it helps.

Hi Bunuel,

My doubt was regarding the distinct factors.

For example

we take 16 -

Wont we consider the negative factors also?

Like for eg - for 16 they would be -1, 1, -2, 2, -4, 4, -8, 8, -16, 16 so that gives us an even number of distinct factors right?

Why wont we consider the negative in this case. The integers/factors with the negative sign are distinct too.

Will be grateful if you could clarify a little.

Thanks

Please read the red part in the post you quote: factor is a "positive divisor" (at least on the GMAT).
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15413
Followers: 649

Kudos [?]: 206 [0], given: 0

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

22 May 2015, 12:11
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.
_________________
Optimus Prep Instructor
Joined: 06 Nov 2014
Posts: 1812
Followers: 55

Kudos [?]: 441 [0], given: 22

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

27 May 2015, 14:14
Which of the following must be true if the square root of X is a positive integer?
This tells us that X is a perfect square. It will therefore not have an even number of distinct factors and we can eliminate A, C, and E.
The difference between B and D is choice III so we evaluate that. Since perfect squares always have an odd number of distinct factors, the sum of the distinct factors will be odd. That leaves only choice D.

I. X has an even number of distinct factors.
II. X has an odd number of distinct factors.
III. The sum of X’s distinct factors is odd.

(A) I only
(B) II only
(C) I and III
(D) II and III
(E) I, II, and III
_________________

# Janielle Williams

Customer Support

Special Offer: $80-100/hr. Online Private Tutoring GMAT On Demand Course$299
Free Online Trial Hour

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15413
Followers: 649

Kudos [?]: 206 [0], given: 0

Re: Which of the following must be true if the square root of X [#permalink]

### Show Tags

06 Jun 2016, 12:04
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.
_________________
Re: Which of the following must be true if the square root of X   [#permalink] 06 Jun 2016, 12:04
Similar topics Replies Last post
Similar
Topics:
5 If √x=x , then which of the following must be true ? 3 29 Mar 2016, 06:20
20 If |x|=−x, which of the following must be true? 7 04 Apr 2017, 07:29
8 If √x = x, then which of the following must be true? 4 11 Nov 2015, 12:32
3 If x-y=8, which of the following must be true? 5 20 Apr 2016, 23:55
29 If x/|x|, which of the following must be true for all 14 30 Jun 2016, 10:48
Display posts from previous: Sort by

# Which of the following must be true if the square root of X

 Powered by phpBB © phpBB Group and phpBB SEO 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®.