GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 25 May 2020, 07:43 ### 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.  # Question of the Week- 11 ( If p is positive integer and p^2 has ..)

Author Message
TAGS:

### Hide Tags

e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3367
Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

13 00:00

Difficulty:   65% (hard)

Question Stats: 56% (02:14) correct 44% (02:16) wrong based on 111 sessions

### HideShow timer Statistics

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

_________________

Originally posted by EgmatQuantExpert on 11 Aug 2018, 09:57.
Last edited by EgmatQuantExpert on 12 Aug 2018, 23:17, edited 2 times in total.
Math Expert V
Joined: 02 Aug 2009
Posts: 8587
Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

2
1
EgmatQuantExpert wrote:
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
_________________
Director  P
Joined: 14 Dec 2017
Posts: 502
Location: India
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

1
EgmatQuantExpert wrote:
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 $$p = 25 - 9 = 16$$

Thanks,
GyM
_________________
Senior Manager  P
Joined: 31 Jul 2017
Posts: 499
Location: Malaysia
GMAT 1: 700 Q50 V33
GPA: 3.95
WE: Consulting (Energy and Utilities)
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

EgmatQuantExpert wrote:
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.
Intern  S
Joined: 10 Feb 2017
Posts: 42
Location: Viet Nam
GPA: 3.5
WE: General Management (Education)
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

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
Senior Manager  P
Joined: 31 Jul 2017
Posts: 499
Location: Malaysia
GMAT 1: 700 Q50 V33
GPA: 3.95
WE: Consulting (Energy and Utilities)
Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

1
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.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3367
Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

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.

_________________
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3367
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

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

_________________
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3367
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#permalink]

### Show Tags

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."
_________________
Non-Human User Joined: 09 Sep 2013
Posts: 14970
Re: Question of the Week- 11 ( If p is positive integer and p^2 has ..)  [#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: Question of the Week- 11 ( If p is positive integer and p^2 has ..)   [#permalink] 05 May 2020, 21:52

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