NEW HERE? LEARN MORE
closeclose
Last visit was: 24 Apr 2024, 09:01 It is currently 24 Apr 2024, 09:01
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

Question of the Week- 11 ( If p is positive integer and p^2 has ..)

SORT BY:
Date
Kudos
Tags:
Find Similar Topics 
Show Tags
Hide Tags
User avatar
L
EgmatQuantExpert
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3726
Own Kudos [?]: 16832 [23]
Given Kudos: 165
Send PM
Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   Updated on: 13 Aug 2018, 00:17
1
Kudos
22
Bookmarks
Expert Reply
00:00
A
B
C
D
E

Difficulty:

65% (hard)

Question Stats:

62% (02:13) correct 38% (02:12) wrong based on 211 sessions
Hide Show timer Statistics
e-GMAT Question of the Week #11


If \(p\) is a positive integer and \(p^2\) has total \(17\) positive factors, then find the number of positive integers that completely divides \(p^3\) but does not completely divide \(p\)?

Options

(A) 16
(B) 17
(C) 21
(D) 23
(E) 24

To access all the questions: Question of the Week: Consolidated List

ShowHide Answer
Official Answer
A

Originally posted by EgmatQuantExpert on 11 Aug 2018, 10:57.
Last edited by EgmatQuantExpert on 13 Aug 2018, 00:17, edited 2 times in total.
User avatar
L
chetan2u
RC & DI Moderator
Joined: 02 Aug 2009
Status:Math and DI Expert
Posts: 11169
Own Kudos [?]: 31886 [3]
Given Kudos: 290
Send PM
Reviews Badge
Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   11 Aug 2018, 19:57
2
Kudos
1
Bookmarks
Expert Reply
EgmatQuantExpert wrote:
e-GMAT Question of the Week #11


If \(p\) is a positive integer and \(p^2\) has total \(17\) positive factors, then find the number of positive integers that completely divides \(p^3\) but does not completely divide \(p\)?

Options

(A) 16
(B) 17
(C) 21
(D) 23
(E) 24

To access all the QOW questions: Question of the Week: Consolidated List



Number of factors =\(17=1*17=(16+1)\),
therefore \(p^2=a^16\) and \(p=a^8\), so p will have \((8+1) =9\) factors
Now \(p^3=(a^8)^3=a^24\), and factors are \((24+1)=25\)

Thus factors that do not divide p but divide \(p^3\) are 2\(5-9=16\)

A
User avatar
P
GyMrAT
Senior Manager
Senior Manager
Joined: 14 Dec 2017
Posts: 426
Own Kudos [?]: 459 [1]
Given Kudos: 173
Location: India
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   11 Aug 2018, 22:33
1
Kudos
EgmatQuantExpert wrote:
e-GMAT Question of the Week #11


If \(p\) is a positive integer and \(p^2\) has total \(17\) positive factors, then find the number of positive integers that completely divides \(p^3\) but does not completely divide \(p\)?

Options

(A) 16
(B) 17
(C) 21
(D) 23
(E) 24




Given \(p > 0\) & \(p^2\) has 17 factors, hence \(p\) is of the form \(n^8\) & has \((8+1) = 9\) factors

we get \(p^2 = (n^{8})^{2} = n^{16}\)

& \(p^3 = (n^{8})^{3} = n^{24}\), which has \((24+1)\) = \(25\) factors

Therefore # of factors of p^3 that are not factors of \(p = 25 - 9 = 16\)


Answer A.


Thanks,
GyM
User avatar
P
rahul16singh28
Senior Manager
Senior Manager
Joined: 31 Jul 2017
Posts: 435
Own Kudos [?]: 443 [0]
Given Kudos: 752
Location: Malaysia
Schools: INSEAD Jan '19
GPA: 3.95
WE:Consulting (Energy and Utilities)
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   12 Aug 2018, 07:19
EgmatQuantExpert wrote:
e-GMAT Question of the Week #11


If \(p\) is a positive integer and \(p^2\) has total \(17\) positive factors, then find the number of positive integers that completely divides \(p^3\) but does not completely divide \(p\)?

Options

(A) 16
(B) 17
(C) 21
(D) 23
(E) 24

To access all the QOW questions: Question of the Week: Consolidated List



As \(p^2\) has 17 factors, so \(p^2 = 2^{16}\) or \(p = 2^8\)........ (p can be any Prime Number)
Now, \(p^3 = 2^{24}\).. As per the question statement, the dividing Integer should divide \(2^{24}\) but not \(2^8\)...... So, the dividing Integer will be \(2^9, 2^{10},..... 2^{24}.\)

Hence, A.
User avatar
G
Hungluu92vn
Intern
Intern
Joined: 10 Feb 2017
Posts: 34
Own Kudos [?]: 83 [0]
Given Kudos: 163
Location: Viet Nam
GPA: 3.5
WE:General Management (Education)
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   14 Aug 2018, 22:55
Can someone help me to elaborate more why you can conclude p^2 has 17 factors so p^2 = 2^16 or a^16. Sorry i dont get this point. Thank you so much
User avatar
P
rahul16singh28
Senior Manager
Senior Manager
Joined: 31 Jul 2017
Posts: 435
Own Kudos [?]: 443 [1]
Given Kudos: 752
Location: Malaysia
Schools: INSEAD Jan '19
GPA: 3.95
WE:Consulting (Energy and Utilities)
Send PM
Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   14 Aug 2018, 23:03
1
Kudos
Hungluu92vn wrote:
Can someone help me to elaborate more why you can conclude p^2 has 17 factors so p^2 = 2^16 or a^16. Sorry i dont get this point. Thank you so much


Total Number of Factors for \(a^{n} = n+1\).. Hope this is clear.
User avatar
L
EgmatQuantExpert
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3726
Own Kudos [?]: 16832 [2]
Given Kudos: 165
Send PM
Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   16 Aug 2018, 00:55
2
Bookmarks
Expert Reply

Solution



Given:
    • p is a positive integer
    • \(p^2\) has 17 positive factors

To find:
    • Number of positive integers that divides \(p^3\), but does not divide p

Approach and Working:
    • The number of factors of p^2 is 17
      o 17 is a prime number, so it can only be written as 1 * 17 = (0+1) * (16+1)
      o Thus, \(p^2 = P_1^{16}\), where \(P_1\) is a prime number
    • Implies, \(p = P_1^8\)
      o The number of factors of p will be 9
    • And, \(p^3 = P_1^{24}\)
      o The number of factors of \(p^3\) will be 25
    • All the factors of ‘p’ will also be the factors of \(p^3\)

Therefore, the number of positive integers that divides p^3, but does not divide p = 25 – 9 = 16

Hence, the correct answer is option A.

Answer: A

User avatar
L
EgmatQuantExpert
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3726
Own Kudos [?]: 16832 [1]
Given Kudos: 165
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   16 Aug 2018, 01:14
1
Bookmarks
Expert Reply
Hungluu92vn wrote:
Can someone help me to elaborate more why you can conclude p^2 has 17 factors so p^2 = 2^16 or a^16. Sorry i dont get this point. Thank you so much


Since, \(p^2 = 17\), is a prime number,
• It can only be written as 1*17 = (0+1) * (16+1)
• And, we know that the number of factors of \(N = p^a * q^b * r^c\)... is (a+1) * (b+1) * (c+1) ...
Thus, \(p^2 = P_1^{16}\), where \(P_1\) is a prime number

User avatar
L
EgmatQuantExpert
e-GMAT Representative
Joined: 04 Jan 2015
Posts: 3726
Own Kudos [?]: 16832 [0]
Given Kudos: 165
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   16 Aug 2018, 01:22
Expert Reply
rahul16singh28 wrote:

As \(p^2\) has 17 factors, so \(p^2 = 2^{16}\) or \(p = 2^8\)........ (p can be any Prime Number)
Now, \(p^3 = 2^{24}\).. As per the question statement, the dividing Integer should divide \(2^{24}\) but not \(2^8\)...... So, the dividing Integer will be \(2^9, 2^{10},..... 2^{24}.\)

Hence, A.


There is a small correction in your solution.

The statement "p can be any Prime Number" is incorrect, because p is not a prime number,

Instead, it should have been stated as "p can have any prime number/factor in place of 2."
User avatar
bumpbot
Non-Human User
Joined: 09 Sep 2013
Posts: 32648
Own Kudos [?]: 821 [0]
Given Kudos: 0
Send PM
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink] New post   16 Oct 2023, 12:56
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 Bot
gmatclubot
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..) [#permalink]
16 Oct 2023, 12:56
Moderators:
Bunuel
Math Expert
92902 posts
yashikaaggarwal
Senior Moderator - Masters Forum
3137 posts

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