It is currently 26 Jun 2017, 10:59

### 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 x is an integer that has exactly three positive divisors

Author Message
TAGS:

### Hide Tags

Intern
Joined: 18 Feb 2010
Posts: 28
If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

16 May 2012, 20:10
4
This post was
BOOKMARKED
00:00

Difficulty:

35% (medium)

Question Stats:

61% (01:54) correct 39% (00:52) wrong based on 243 sessions

### HideShow timer Statistics

If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?

A. 4
B. 5
C. 6
D. 7
E. 8

I am not sure how the answer is derived here
If x=2, then X^3 =8 and 8 has 4 divisors - 1,2,4,8
But if x=9, then 9^3 =3^6, will have 7 divisors. So isn't the number of positive divisors dependent on the value of x?

Thanks
[Reveal] Spoiler: OA
Manager
Joined: 28 May 2011
Posts: 193
Location: United States
GMAT 1: 720 Q49 V38
GPA: 3.6
WE: Project Management (Computer Software)
Re: Number of positive divisiors of X^3 [#permalink]

### Show Tags

16 May 2012, 21:33
If x is an integer that has exactly three positive divisors (1, x, y)
=> It means x is a square number with x = y^2
So possible divisors of X^3 could be :
1, y, y^2 ---- (Divisors of x)
y^3, y^4 ---- (Additional Divisors of x^2)
y^5, y^6 ---- (Additional Divisors of x^3)

Regarding nbr 2 => It has only 2 divisors and those are 1 and 2.

Hope it clarifies situation.
_________________

-------------------------------------------------------------------------------------------------------------------------------
http://gmatclub.com/forum/a-guide-to-the-official-guide-13-for-gmat-review-134210.html
-------------------------------------------------------------------------------------------------------------------------------

Math Expert
Joined: 02 Sep 2009
Posts: 39701
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

17 May 2012, 01:10
2
KUDOS
Expert's post
2
This post was
BOOKMARKED
If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?

A. 4
B. 5
C. 6
D. 7
E. 8

I am not sure how the answer is derived here
If x=2, then X^3 =8 and 8 has 4 divisors - 1,2,4,8
But if x=9, then 9^3 =3^6, will have 7 divisors. So isn't the number of positive divisors dependent on the value of x?

Thanks

x cannot be 2, because 2 has only two divisors 1 and 2, not three as given in the stem.

If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?
A. 4
B. 5
C. 6
D. 7
E. 8

Important property: 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. (A perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square).

Hence, since given that x has 3 (odd) divisors then x is a perfect square, specifically square of a prime. The divisor of $$x$$ are: $$1$$, $$\sqrt{x}=prime$$ and $$x$$ itself. So, $$x$$ can be 4, 9, 25, ... For example divisors of 4 are: 1, 2=prime, and 4 itself.

Now, $$x^3=(\sqrt{x})^6=prime^6$$, so it has 6+1=7 factors (check below for that formula).

Else you can just plug some possible values for $$x$$: say $$x=4$$ then $$x^3=64=2^6$$ --> # of factors of 2^6 is 6+1=7.

Finding the Number of Factors of an Integer

First make prime factorization of an integer $$n=a^p*b^q*c^r$$, where $$a$$, $$b$$, and $$c$$ are prime factors of $$n$$ and $$p$$, $$q$$, and $$r$$ are their powers.

The number of factors of $$n$$ will be expressed by the formula $$(p+1)(q+1)(r+1)$$. NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: $$450=2^1*3^2*5^2$$

Total number of factors of 450 including 1 and 450 itself is $$(1+1)*(2+1)*(2+1)=2*3*3=18$$ factors.

So, the # of factors of x=a^2*b^3, where a and b are different prime numbers is (2+1)(3+1)=12.

Hope it's clear.
_________________
Intern
Joined: 18 Feb 2010
Posts: 28
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

17 May 2012, 13:56
Thanks very much both of you for the responses.
Although I could get that x is a perfect square, since it has 3 factors including itself, I think I failed to correlate x = y^2.
Once that is done, I think x= y^6 and the number of positive divisors will be 7.

Thanks again for the explanations!
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15982
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

30 Sep 2013, 04:11
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.
_________________
Director
Joined: 03 Aug 2012
Posts: 893
Concentration: General Management, General Management
GMAT 1: 630 Q47 V29
GMAT 2: 680 Q50 V32
GPA: 3.7
WE: Information Technology (Investment Banking)
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

30 Sep 2013, 11:52
Hit and trial one such number we have '4'

Since factors of '4'are {1,2,4}

4^3=64

Number of factors/divisors of 64= 2^6

We know that when a number is expressed as a product of the prime factors as below:

N = a^x * b^y * c^z

Then no. of divisors = (x+1)*(y+1)*(z+1)

Then here (6+1) = 7

!
_________________

Rgds,
TGC!
_____________________________________________________________________
I Assisted You => KUDOS Please
_____________________________________________________________________________

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 15982
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

13 Dec 2014, 11:33
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 Legend
Joined: 09 Sep 2013
Posts: 15982
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

04 Jun 2016, 15:10
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.
_________________
Manager
Joined: 12 Jan 2016
Posts: 72
Location: United States
Concentration: Operations, General Management
GMAT 1: 690 Q49 V35
GPA: 3.5
WE: Supply Chain Management (Consumer Electronics)
If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

04 Jun 2016, 20:54
X has odd number of factors, so x is a perfect square. Once you know that x is a perfect square, you know that $$x^3$$ is also a perfect square. Perfect squares have odd number of factors and in the options, only Option D is odd. So, answer is D.
Math Forum Moderator
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 2893
Location: India
GPA: 3.5
Re: If x is an integer that has exactly three positive divisors [#permalink]

### Show Tags

05 Jun 2016, 00:19
If x is an integer that has exactly three positive divisors (these include 1 and x), how many positive divisors does x^3 have?

A. 4
B. 5
C. 6
D. 7
E. 8

I am not sure how the answer is derived here
If x=2, then X^3 =8 and 8 has 4 divisors - 1,2,4,8
But if x=9, then 9^3 =3^6, will have 7 divisors. So isn't the number of positive divisors dependent on the value of x?

Thanks

Consider simple numbers like 4

4 has 3 divisors, 1 , 2 and 4

Now 4^3 = 64

Factors of 64 are 1, 2, 4, 8, 16, 32, 64

So, THere are 7 factors of x^3

Hope this helps...

_________________

Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )

Re: If x is an integer that has exactly three positive divisors   [#permalink] 05 Jun 2016, 00:19
Similar topics Replies Last post
Similar
Topics:
10 x and y are positive integers. If the greatest common divisor of x and 5 03 Feb 2017, 19:04
2 The positive integers p and r have exactly three prime factors in 6 17 Oct 2015, 07:37
15 A positive integer x has 60 divisors and 7x has 80 divisors. What is t 12 31 Oct 2016, 10:06
21 The three-digit positive integer x has the hundreds, 11 28 Apr 2017, 04:57
25 If the integer n has exactly three positive divisors, includ 11 18 Apr 2017, 16:14
Display posts from previous: Sort by