If the square root of p^2 is an integer, which of the follow : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 20 Jan 2017, 07:09

### 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

# If the square root of p^2 is an integer, which of the follow

Author Message
TAGS:

### Hide Tags

Manager
Joined: 14 Mar 2010
Posts: 82
Followers: 2

Kudos [?]: 135 [2] , given: 44

If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

19 Jan 2013, 20:31
2
KUDOS
3
This post was
BOOKMARKED
00:00

Difficulty:

65% (hard)

Question Stats:

47% (01:38) correct 53% (00:43) wrong based on 242 sessions

### HideShow timer Statistics

If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III
[Reveal] Spoiler: OA

_________________

MGMAT CAT MATH http://gmatclub.com/forum/mgmat-cat-math-144609.html
MGMAT SC SUMMARY: http://gmatclub.com/forum/mgmat-sc-summary-144610.html

Last edited by monir6000 on 20 Jan 2013, 19:41, edited 2 times in total.
Manager
Joined: 12 Mar 2012
Posts: 94
Location: India
Concentration: Technology, Strategy
GMAT 1: 710 Q49 V36
GPA: 3.2
WE: Information Technology (Computer Software)
Followers: 9

Kudos [?]: 317 [0], given: 22

Re: If the square root of p^2 is an integer [#permalink]

### Show Tags

19 Jan 2013, 20:48
Answer should be D and not B.
Every square has odd number of factors.
Manager
Status: Never ever give up on yourself.Period.
Joined: 23 Aug 2012
Posts: 152
Location: India
Concentration: Finance, Human Resources
GMAT 1: 570 Q47 V21
GMAT 2: 690 Q50 V33
GPA: 3.5
WE: Information Technology (Investment Banking)
Followers: 10

Kudos [?]: 302 [0], given: 35

Re: If the square root of p^2 is an integer [#permalink]

### Show Tags

19 Jan 2013, 22:40
Yes, D has to be the answer.
_________________

Don't give up on yourself ever. Period.
Beat it, no one wants to be defeated (My journey from 570 to 690) : http://gmatclub.com/forum/beat-it-no-one-wants-to-be-defeated-journey-570-to-149968.html

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 13461
Followers: 575

Kudos [?]: 163 [0], given: 0

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

07 Mar 2014, 16:53
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.
_________________
Senior Manager
Joined: 20 Dec 2013
Posts: 273
Location: India
Followers: 0

Kudos [?]: 81 [0], given: 29

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

07 Mar 2014, 17:10
The first and third statements are clear.
A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no.
Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true?
Is repetition of a prime factor counted in calculating even no. Of prime factors?

Posted from my mobile device
Math Expert
Joined: 02 Sep 2009
Posts: 36582
Followers: 7086

Kudos [?]: 93247 [6] , given: 10555

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

08 Mar 2014, 05:29
6
KUDOS
Expert's post
4
This post was
BOOKMARKED
AKG1593 wrote:
If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III

The first and third statements are clear.
A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no.
Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true?
Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case:
The square root of p^2 is an integer --> $$\sqrt{p^2}=integer$$ --> $$p=integer$$.

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Hope it helps.
_________________
Intern
Joined: 25 Feb 2014
Posts: 12
Followers: 0

Kudos [?]: 3 [0], given: 2

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

01 Jun 2014, 23:09
Bunuel wrote:
AKG1593 wrote:
If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III

The first and third statements are clear.
A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no.
Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true?
Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case:
The square root of p^2 is an integer --> $$\sqrt{p^2}=integer$$ --> $$p=integer$$.

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Hope it helps.

Hi Bunell,

Can you please tell whether while counting the total number of factors of a perfect square, do we count 1 and the number itself?

For example :- for any number lets say 6. the total number of factors will be 1,2,3,6 .

Isn't the case with perfect squares? as if we include 1 and number itself , the total number of factors of a perfect square will be even . like for 36- 1,2,3,4,6,9,18,36

Please let me kow what i am approaching wrong
Manager
Joined: 20 Dec 2013
Posts: 127
Followers: 10

Kudos [?]: 93 [0], given: 1

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

01 Jun 2014, 23:52
monir6000 wrote:
If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III

I. p^2 is a perfect square and a perfect square has pairs of factors and '1' with it. The total number of factors is odd
II. If p = 3 then p^2 = 9. The prime factors will always remain even as the p is an integer and the pairs have to come out of square root to make p an integer.
III. This is not true always. Let us say p = 4, it has three factors: 1, 2 and 4. Hence it will not hold true always.

_________________

76000 Subscribers, 7 million minutes of learning delivered and 5.6 million video views

Perfect Scores
http://perfectscores.org

Math Expert
Joined: 02 Sep 2009
Posts: 36582
Followers: 7086

Kudos [?]: 93247 [0], given: 10555

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

02 Jun 2014, 00:04
Manik12345 wrote:
Bunuel wrote:
AKG1593 wrote:
If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III

The first and third statements are clear.
A perfect square will always have odd no. Of factors since one of the factors multiplies with itself to make that no.
Also p can be a square itself so it'll give odd no. Of factors.Not necessarily even no. Of factors.

In second statement,can it be said that because 4 can be written as 2*2(even no. Of prime factors) the statement is true?
Is repetition of a prime factor counted in calculating even no. Of prime factors?

This is a flawed question. The answer to the question cannot be D.

If p=0, then none of the statements must be true.

In order for the answer to be D, the question must specify that p is a positive integer greater than 1.

In this case:
The square root of p^2 is an integer --> $$\sqrt{p^2}=integer$$ --> $$p=integer$$.

I. p^2 has an odd number of factors --> since p is an integer, then p^2 is a perfect square. The number of factors of a positive perfect square is always odd. Thus this option must be true.

II. p^2 can be expressed as the product of an even number of prime factors. Any positive perfect square can be expressed as the product of an even number of prime factors: 4=2*2, 9=3*3, 16=2*2*2*2, 25=5*5, ... each is written as the product of even number of prime factors. Thus this option must be true.

III. p has an even number of factors --> if p itself is a perfect square, 4, 9, ... then this statement won't be true. Discard.

Hope it helps.

Hi Bunell,

Can you please tell whether while counting the total number of factors of a perfect square, do we count 1 and the number itself?

For example :- for any number lets say 6. the total number of factors will be 1,2,3,6 .

Isn't the case with perfect squares? as if we include 1 and number itself , the total number of factors of a perfect square will be even . like for 36- 1,2,3,4,6,9,18,36

Please let me kow what i am approaching wrong

The total number of factors of any positive integer includes 1 and that integer itself. Why there should be an exception for perfect squares? Isn't a perfect square divisible by 1 and itself?

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36 --> 9 factors.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 36582
Followers: 7086

Kudos [?]: 93247 [0], given: 10555

Re: If the square root of p^2 is an integer, which of the follow [#permalink]

### Show Tags

02 Jun 2014, 00:06
PerfectScores wrote:
monir6000 wrote:
If the square root of p^2 is an integer, which of the following must be true?

I. p^2 has an odd number of factors
II. p^2 can be expressed as the product of an even number of prime factors
III. p has an even number of factors

A. I
B. II
C. III
D. I and II
E. II and III

I. p^2 is a perfect square and a perfect square has pairs of factors and '1' with it. The total number of factors is odd
II. If p = 3 then p^2 = 9. The prime factors will always remain even as the p is an integer and the pairs have to come out of square root to make p an integer.
III. This is not true always. Let us say p = 4, it has three factors: 1, 2 and 4. Hence it will not hold true always.

This would be correct if we were told that p is a positive integer greater than 1. Check here: if-the-square-root-of-p-2-is-an-integer-which-of-the-follow-146066.html#p1341427
_________________
Re: If the square root of p^2 is an integer, which of the follow   [#permalink] 02 Jun 2014, 00:06
Similar topics Replies Last post
Similar
Topics:
The square root of 636 is between which set of integers? 2 31 Jul 2016, 03:27
2 If the square root of p^2 is an integer greater than 1, which of the 2 04 Apr 2016, 06:51
3 Which of the following integers is the square of an integer? 7 12 Jun 2015, 02:12
4 Which of the following cannot be the square of an integer 3 05 Sep 2014, 20:47
26 Which of the following must be true if the square root of X 14 21 Dec 2012, 02:44
Display posts from previous: Sort by