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

 It is currently 16 Oct 2019, 05:22 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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  If the integer n has exactly three positive divisors, includ

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

Hide Tags

Intern  Joined: 24 Dec 2012
Posts: 11
Location: United States
Concentration: Finance, Entrepreneurship
GPA: 3
WE: Corporate Finance (Investment Banking)
If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

22 00:00

Difficulty:   25% (medium)

Question Stats: 72% (01:15) correct 28% (01:11) wrong based on 735 sessions

HideShow timer Statistics

If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

OG 11 #241.

Would someone mind explaining? I'm not satisfied with the explanation in the OG.
Math Expert V
Joined: 02 Sep 2009
Posts: 58386
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

13
20
GMATBeast wrote:
If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

OG 11 #241.

Would someone mind explaining? I'm not satisfied with the explanation in the OG.

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

Now, $$n^2=(\sqrt{n})^4=prime^4$$, so it has 4+1=5 factors (check below for that formula).

Else you can just plug some possible values for $$n$$: say $$n=4$$ then $$n^2=16=2^4$$ --> # of factors of 2^4 is 4+1=5.

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.
_________________
General Discussion
Manager  Status: Helping People Ace the GMAT
Joined: 16 Jan 2013
Posts: 182
Location: United States
Concentration: Finance, Entrepreneurship
GMAT 1: 770 Q50 V46 GPA: 3.1
WE: Consulting (Consulting)
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

Basically, the description says that this is the square of a prime number. So if you square that number, you will have a prime number raised to the fourth power.

That will have 5 factors. For a more detailed description, we have a free factors and multiples lesson on our site.
_________________
Want to Ace the GMAT with 1 button? Start Here:
GMAT Answers is an adaptive learning platform that will help you understand exactly what you need to do to get the score that you want.
Intern  Joined: 16 Jun 2012
Posts: 2
Concentration: General Management, Finance
GPA: 3.22
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

quite simple..
take the example of 4...
it has 3 positive divisors (1,2,4)

Now, take the example of 16...
it has only 5 divisors..
so B is the ans
Director  S
Joined: 09 Jun 2010
Posts: 715
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

gmat dose not requires us to remember much.

pick some numbers and see that the number must be a square of prime.

from this departure, we can infer B.

hard one
Manager  Joined: 17 Mar 2014
Posts: 122
Location: United States
GPA: 3.97
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

_________________
KUDOS!!!, I need them too Intern  Joined: 08 Apr 2017
Posts: 1
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

It is 5. (b). If you try and experiment it, any number with three factors total, you will get 4 as one possibility as it is the only possibility there is. so you know that integer n=4. So 2 to the power of 4 is 16. 16 has 5 factors including itself and 1!
VP  P
Joined: 07 Dec 2014
Posts: 1222
If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

GMATBeast wrote:
If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

because n will always be the square of a prime,
positive divisors of n^2 will always be 1, √n, n, n√n, n^2
5
B
Target Test Prep Representative D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8069
Location: United States (CA)
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

1
GMATBeast wrote:
If the integer n has exactly three positive divisors, including 1 and n, how many positive divisors does n^2 have?

(A) 4
(B) 5
(C) 6
(D) 8
(E) 9

Since n has exactly 3 positive divisors we can conclude that n is a perfect square of a prime number. For instance, let’s consider the prime number 3. Notice that 3^2 = 9, and the factors of 9 are 1, 3, and 9.

Thus, if we let n = 9, then 9^2 = 81.

The factors of 81 are 1, 81, 9, 27, and 3. Thus, n^2 has 5 positive divisors.

_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager  B
Joined: 30 Apr 2013
Posts: 76
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

Hi Scott,

Thanks for the explanation but I am bit lost here. In your explaination n^2 has three factors i.e 9 ( 1,3,9) but the question says n has three factors i.e 3 but three has only two factors ( 1,3). Where am I going wrong? Can you also explain the same with one more example?
Math Expert V
Joined: 02 Sep 2009
Posts: 58386
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

santro789 wrote:
Hi Scott,

Thanks for the explanation but I am bit lost here. In your explaination n^2 has three factors i.e 9 ( 1,3,9) but the question says n has three factors i.e 3 but three has only two factors ( 1,3). Where am I going wrong? Can you also explain the same with one more example?

You should read a question and the solutions MUCH more carefully.

In Scott's example, n = 9 NOT n^2.

n = 9 has three factors: 1, 3, and 9. n^2 in this case will be n^2 = 9^2 = 3^4 and it will have 4 + 1 = 5 factors.
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 13174
Re: If the integer n has exactly three positive divisors, includ  [#permalink]

Show Tags

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: If the integer n has exactly three positive divisors, includ   [#permalink] 17 Oct 2018, 02:18
Display posts from previous: Sort by

If the integer n has exactly three positive divisors, includ

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

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