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

 It is currently 17 Nov 2018, 13:44

# Join here

### 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 November
PrevNext
SuMoTuWeThFrSa
28293031123
45678910
11121314151617
18192021222324
2526272829301
Open Detailed Calendar
• ### FREE Quant Workshop by e-GMAT!

November 18, 2018

November 18, 2018

07:00 AM PST

09:00 AM PST

Get personalized insights on how to achieve your Target Quant Score. November 18th, 7 AM PST
• ### How to QUICKLY Solve GMAT Questions - GMAT Club Chat

November 20, 2018

November 20, 2018

09:00 AM PST

10:00 AM PST

The reward for signing up with the registration form and attending the chat is: 6 free examPAL quizzes to practice your new skills after the chat.

# M30-12

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

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

16 Sep 2014, 00:45
3
15
00:00

Difficulty:

85% (hard)

Question Stats:

50% (01:58) correct 50% (01:31) wrong based on 112 sessions

### HideShow timer Statistics

The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?

A. $$0$$
B. $$\frac{1}{2}$$
C. $$1$$
D. $$\frac{3}{2}$$
E. $$2$$

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

16 Sep 2014, 00:45
2
2
Official Solution:

The product of two negative integers, $$a$$ and $$b$$, is a prime number $$p$$. If $$p$$ is the number of factors of $$n$$, where $$n$$ is NOT a perfect square, what is the value of the median of the four integers $$a$$, $$b$$, $$p$$, and $$n$$?

A. $$0$$
B. $$\frac{1}{2}$$
C. $$1$$
D. $$\frac{3}{2}$$
E. $$2$$

This is a hard questions which tests several number theory concepts.

Start from $$n$$: we are told that $$n$$ is NOT a perfect square. The number of factors of perfect square is odd and all other positive integers have even number of factors. Hence, since $$p$$ is the number of factors of $$n$$, then $$p$$ must be even. We also know that $$p$$ is a prime number and since the only even prime is 2, then $$p=2$$. Notice here that from this it follows that $$n$$ must also be a prime, because only primes have 2 factors: 1 and itself.

Next, $$ab=p=2$$ implies that $$a=-1$$ and $$b=-2$$ or vise-versa.

So, the set is {-2, -1, 2, some prime}, which means that the median is $$\frac{-1 + 2}{2} = \frac{1}{2}$$.

_________________
Manager
Joined: 13 Sep 2014
Posts: 88
WE: Engineering (Consulting)

### Show Tags

10 Jan 2015, 23:55
I think this question is good and helpful.
Nice question covering 2-3 concepts
Intern
Joined: 24 Jul 2009
Posts: 2
Concentration: International Business, Technology
GMAT 1: 700 Q49 V38

### Show Tags

08 Apr 2015, 06:34
if the set is {-1,-2,2,some prime} then the median would be 0. Am I missing out something?
Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

08 Apr 2015, 06:39
sachin2040 wrote:
if the set is {-1,-2,2,some prime} then the median would be 0. Am I missing out something?

The median of a set with even number of elements is the average of two middle terms when arranged in ascending /descending order. The set in ascending order is {-2, -1, 2, some prime} --> median = (-1 + 2)/2 = 1/2.
_________________
Intern
Joined: 24 Jul 2009
Posts: 2
Concentration: International Business, Technology
GMAT 1: 700 Q49 V38

### Show Tags

08 Apr 2015, 06:44
Thanks Bunuel, that was silly on my part.....
Retired Moderator
Joined: 23 Sep 2015
Posts: 382
Location: France
GMAT 1: 690 Q47 V38
GMAT 2: 700 Q48 V38
WE: Real Estate (Mutual Funds and Brokerage)

### Show Tags

28 Apr 2016, 06:37
great concept question!
_________________

New Application Tracker : update your school profiles instantly!

Intern
Joined: 09 Nov 2015
Posts: 2

### Show Tags

28 May 2016, 22:04
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.
Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

30 May 2016, 13:18
1
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.
_________________
Senior Manager
Joined: 08 Jun 2015
Posts: 435
Location: India
GMAT 1: 640 Q48 V29
GMAT 2: 700 Q48 V38
GPA: 3.33

### Show Tags

26 Jun 2016, 05:51
A very good question ! +1 Kudos
_________________

" The few , the fearless "

Manager
Joined: 23 Apr 2014
Posts: 63
Location: United States
GMAT 1: 680 Q50 V31
GPA: 2.75

### Show Tags

01 Aug 2016, 13:09
I think this is a high-quality question and I agree with explanation.
Senior Manager
Joined: 12 Mar 2013
Posts: 251

### Show Tags

04 Aug 2016, 16:25
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Then, Suppose, 3 is the number of factors of N.
Can we say, N is a perfect Square?
_________________

We Shall Overcome... One day...

Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

05 Aug 2016, 03:30
nahid78 wrote:
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Then, Suppose, 3 is the number of factors of N.
Can we say, N is a perfect Square?

Yes.

Tips about the perfect square:
1. 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;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: $$36=2^2*3^2$$, powers of prime factors 2 and 3 are even.

Hope it helps.
_________________
Manager
Joined: 28 Sep 2013
Posts: 86
GMAT 1: 740 Q51 V39

### Show Tags

06 Oct 2016, 01:29
Bunuel wrote:
So, the set is {-2, -1, 2, some prime}, which means that the median is $$\frac{-1 + 2}{2} = \frac{1}{2}$$. Answer: B

Didn't Get It!
_________________

Richa Champion | My GMAT Journey - 470 720 740

Target 760+

Not Improving after Multiple attempts. I can guide You.
Contact me richacrunch2@gmail.com

Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

06 Oct 2016, 03:53
RichaChampion wrote:
Bunuel wrote:
So, the set is {-2, -1, 2, some prime}, which means that the median is $$\frac{-1 + 2}{2} = \frac{1}{2}$$. Answer: B

Didn't Get It!

The median is the average of two middle terms, when arranged in ascending/descending order. So, the median of {-2, -1, 2, some prime} is $$\frac{-1 + 2}{2} = \frac{1}{2}$$ (regardless of the unknown prime there).
_________________
Intern
Joined: 14 Apr 2016
Posts: 1

### Show Tags

17 Nov 2016, 14:41
Bunuel wrote:
sushruthav wrote:
Notice here that from this it follows that n must also be a prime, because only primes have 2 factors: 1 and itself.

Can n be negative here? Say n = -1, its factors would be 1 and -1.

No, a factor is a positive divisor.

Could you please explain a bit more regarding factors of negative number? Can n be -2 in this case having two factors 1 and 2? From my understanding if number of factors of a number are given the number could be either positive or negative.
Current Student
Joined: 12 Oct 2012
Posts: 114
WE: General Management (Other)

### Show Tags

19 Nov 2016, 00:12
Great Question. Very helpful. +1 Kudos.
Intern
Joined: 03 Nov 2013
Posts: 2
Location: Australia
GMAT 1: 650 Q48 V31

### Show Tags

09 Jan 2017, 19:25
I think this is a high-quality question and I agree with explanation.
Intern
Joined: 29 Jun 2016
Posts: 7
Location: India
Schools: ISB '20 (A)
GMAT 1: 690 Q48 V35
GPA: 3.82

### Show Tags

21 Apr 2018, 03:08
Since a and b can be -1 and -2 or vice versa, are 1/2 and 0 not a possible answer? How did we assume that a=-2 and b=-1?
Math Expert
Joined: 02 Sep 2009
Posts: 50623

### Show Tags

21 Apr 2018, 04:16
prkohli wrote:
Since a and b can be -1 and -2 or vice versa, are 1/2 and 0 not a possible answer? How did we assume that a=-2 and b=-1?

Your question is not clear. How are you getting 0 or 1/2 for the median? What set do you have giving those values of median ?

We got that $$a=-1$$ and $$b=-2$$ or vise-versa.

So, the set is {-2, -1, p = 2, n = some prime}, which means that the median is (-1 + 2)/2 = 1/2.
_________________
Re: M30-12 &nbs [#permalink] 21 Apr 2018, 04:16

Go to page    1   2    Next  [ 28 posts ]

Display posts from previous: Sort by

# M30-12

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

Moderators: chetan2u, Bunuel

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.