NEW HERE? LEARN MORE
closeclose
It is currently 30 Nov 2023, 12:31
Register
GMAT Club Tests
GMAT Club Tests
Decision Tracker
My Rewards
New posts
New comers' posts

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.

Go to My Error Log Learn more
Close

Request Expert Reply

Confirm Cancel

If a positive integer n has k positive factors, 2n has 2k positive

7 posts Go to First Unread Post
Print view
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
SORT BY:
Date
Kudos
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 90114
CAT Tests Forum Quiz Mode
If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 00:50
3
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

95% (hard)

Question Stats:

28% (02:27) correct 72% (02:39) wrong based on 54 sessions

Hide Show timer Statistics

If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
ShowHide Answer
Official Answer
Official Answer and Stats are available only to registered users. Register/Login.

_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
L
GMATinsight
GMAT Club Legend
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 5881
Location: India
GMAT: QUANT EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Reviews Badge
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 07:12
1
Kudos
Expert Reply
Bunuel wrote:
If a positive integer n has k positive factors, 2n has 2k positive factors, and 3n has 6k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

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


If a positive integer n has k positive factors, 2n has 2k positive factors,

i.e. n = ODD. But all numbers here are odd


n = 177 = 3*59
Number of factors of 177 = (1+1)*(1+1) = 4
3n = 3*177 = 3^2*59
Number of factors of 3*177 = (2+1)*(1+1) = 6
so the factors are not 6 times OUT


n = 231 = 3*7*11
Number of factors of 231 = (1+1)*(1+1)*(1+1) = 8
3n = 3*231 = 3^2*7*11
Number of factors of 3*231 = (2+1)*(1+1)*(1+1) = 12
so the factors are not 6 times OUT


n = 279 = 3^2*31
Number of factors of 279 = (2+1)*(1+1) = 6
3n = 3*279 = 3^3*31
Number of factors of 3*279 = (3+1)*(1+1) = 8
so the factors are not 6 times OUT


No value can be n as per second condition.
User avatar
L
Bunuel
Math Expert
Joined: 02 Sep 2009
Posts: 90114
CAT Tests Forum Quiz Mode
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 08:33
Expert Reply
GMATinsight wrote:
Bunuel wrote:
If a positive integer n has k positive factors, 2n has 2k positive factors, and 3n has 6k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

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


If a positive integer n has k positive factors, 2n has 2k positive factors,

i.e. n = ODD. But all numbers here are odd


n = 177 = 3*59
Number of factors of 177 = (1+1)*(1+1) = 4
3n = 3*177 = 3^2*59
Number of factors of 3*177 = (2+1)*(1+1) = 6
so the factors are not 6 times OUT


n = 231 = 3*7*11
Number of factors of 231 = (1+1)*(1+1)*(1+1) = 8
3n = 3*231 = 3^2*7*11
Number of factors of 3*231 = (2+1)*(1+1)*(1+1) = 12
so the factors are not 6 times OUT


n = 279 = 3^2*31
Number of factors of 279 = (2+1)*(1+1) = 6
3n = 3*279 = 3^3*31
Number of factors of 3*279 = (3+1)*(1+1) = 8
so the factors are not 6 times OUT


No value can be n as per second condition.


Sorry, there was a typo in the text. Edited. Can you please try again? Thank you!
_________________
New to the GMAT CLUB Forum?

My Signature Questions' Collection:


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics
Signature Read More
User avatar
B
pratyush3284
Intern
Intern
Joined: 29 Jul 2017
Posts: 14
Location: India
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 10:30
177 = 3^1 x 59^1; Factors = 2 x 2 (add 1 to each power and mulitply) = 4 =k
177 x 2 = 2^1 x 59 ^1 x 3^1 = 2x2x2 = 8 = 2k
177 x 6 = 2^1 x 59^1 x 3^2 = 2x2x3 = 12 = 3k

Similarly 231 also follows the same; however, 279 doesnt.

Therefore, Ans = C
User avatar
L
Archit3110
GMAT Club Legend
GMAT Club Legend
Joined: 18 Aug 2017
Status:You learn more from failure than from success.
Posts: 7812
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE:Marketing (Energy and Utilities)
GMAT ToolKit User Reviews Badge CAT Tests
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 10:40
Use option to solve
n is 177 factors (k)= 3*59;4
2*177 factors 8 i.e. 2k
And 6n factors
2*3^2*59: 12=3k
Sufficient

If n is 231
3*7*11 :8
2*231:16 factors i.e. 2k
And 6n has
2*3^2*7*11 has 24 i.e. 3k
Sufficient

If n is 279
K here is 3^2*31= 6
2k has 2*3^2*31= 12=2k
6k has 2*3^3*31= 16 not equal to 3k
InSufficient

OPTION C IS CORRECT


Bunuel wrote:
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

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


Posted from my mobile device
User avatar
L
gmatophobia
Quant Chat Moderator
Joined: 22 Dec 2016
Posts: 2801
Location: India
Concentration: Strategy, Leadership
CAT Tests Forum Quiz Mode
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  25 Sep 2023, 21:52
Bunuel wrote:
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

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


We can find the number of factors for 2n, and 3n as shown below -

Attachment:
Screenshot 2023-09-26 111948.jpg
Screenshot 2023-09-26 111948.jpg [ 80.5 KiB | Viewed 684 times ]


For I and II the positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors.

Option C
_________________
Want to discuss quant questions and strategies : Join the quant chat group today

Power of Tiny Gains

1.01^(365) = 37.78

0.99^(365) = 0.03
Signature Read More
User avatar
S
coucoudanger
Intern
Intern
Joined: 20 Aug 2023
Posts: 34
Location: Algeria
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink] New post  26 Sep 2023, 05:36
Bunuel wrote:
If a positive integer n has k positive factors, 2n has 2k positive factors, and 6n has 3k positive factors, which of the following can be the value of n?

I. 177
II. 231
III. 279

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


I. 177
\(n=177=59*3\)
k\(=(1+1)*(1+1) = 4\)
\(2n=59*3*2\)
\(k=2*2*2=8=2\)k
\(6n=59*3^{2}*2\)
\(k=2*3*2=12=3\)k

So 177 can be the value of n

II. 231
\(n=231=3*7*11\)
k\(=2*2*2=8\)
\(2n=2*3*7*11\)
\(k=2*2*2*2=16=2\)k
\(6n=2*3^{2}*7*11\)
\(k=2*3*2*2=24=3\)k

231 can be the value of n

III. 279

\(n=279=3^{2}*31\)
k=\(3*2=6\)
\(2n=2*3^{2}*31\)
\(k=2*3*2=12=2\)k
\(6n=2*3^{3}*31\)
\(k=2*4*2=16\) does not equal 3k
279 can not be the value of n

The correct answer is C
GMAT Club Bot
gmatclubot
Re: If a positive integer n has k positive factors, 2n has 2k positive [#permalink]
26 Sep 2023, 05:36
NEW TOPIC
POST REPLY
Downloads
My Bookmarks
Reviews
Moderators:
Bunuel
Math Expert
90114 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3141 posts

cron

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne
Main navigation
GMAT Resources
Partners
MBA Resources
www.gmac.com|www.mba.com | | GMAT Club Rules | Terms and Conditions