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

 It is currently 23 Oct 2018, 11:24

### 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

# How many factors does x have, if x is a positive integer ?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 50058
How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

17 Nov 2014, 12:29
2
5
00:00

Difficulty:

35% (medium)

Question Stats:

68% (01:27) correct 32% (01:23) wrong based on 258 sessions

### HideShow timer Statistics

Tough and Tricky questions: Divisibility/Multiples/Factors.

How many factors does $$x$$ have, if $$x$$ is a positive integer?

(1) $$x = p^n$$, where $$p$$ is a prime number

(2) $$n^n = n + n$$, where $$n$$ is a positive integer

Kudos for a correct solution.

_________________
Manager
Joined: 05 Jun 2014
Posts: 62
GMAT 1: 630 Q42 V35
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

18 Nov 2014, 02:13
2
Statement 1 tells that the no of factors of x is n+1.
Statement 2 is clearly not sufficient, but it tells that n=2.
Together they tell that x has 3 factors. Answer is C.
Manager
Joined: 21 Jan 2014
Posts: 62
WE: General Management (Non-Profit and Government)
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

18 Nov 2014, 07:03
x is a positive integer

from statement 1 : x=p^n, where p is prime
so the no of factors will be n+1.
value of n will lead to complete solution.

from statement 2: n^n=n+n, where n is positive integer
n^n=2n
n^n-2n=0
n(((n^(n-1))-2)=0
hence either n=0 or n^(n-1)=2 that is n=2

combining both statements will yield two different no of factors for two different value of n.

Manager
Joined: 05 Jun 2014
Posts: 62
GMAT 1: 630 Q42 V35
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

18 Nov 2014, 07:33
We are told that n is +ve, you did everything correct but n cant be 0 as n is +ve. So n=2.
Math Expert
Joined: 02 Sep 2009
Posts: 50058
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

18 Nov 2014, 07:48
1
1
Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

How many factors does $$x$$ have, if $$x$$ is a positive integer?

(1) $$x = p^n$$, where $$p$$ is a prime number

(2) $$n^n = n + n$$, where $$n$$ is a positive integer

Kudos for a correct solution.

Official Solution:

How many factors does $$x$$ have, if $$x$$ is a positive integer?

We cannot easily rephrase the question. Note that we may not need to know $$x$$ in order to know how many factors it has.

Statement (1): INSUFFICIENT. Without knowing the value of $$n$$, we cannot determine the number of factors $$x$$ has.

Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable $$n$$, whereas the question only involves the variable $$x$$.

Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of $$n$$.

Since $$n$$ is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.

$$1^1 = 1 + 1$$? No.

$$2^2 = 2 + 2$$? Yes.

$$3^3 = 3 + 3$$? No.

$$4^4 = 4 + 4$$? No.

Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of $$n$$. Thus, we can determine that the value of $$n$$ is 2.

Now, we do not know the value of $$p$$, nor of $$x$$, but we do now know that $$x = p^2$$, with $$p$$ as a prime number. Since a prime number has no factors other than 1 and itself, we can see that $$x$$ has no factors other than 1, $$p$$, and $$p^2$$. Thus, $$x$$ has exactly 3 factors, and we can answer the question definitively.

_________________
Intern
Joined: 23 May 2016
Posts: 10
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

31 May 2016, 18:00
Bunuel wrote:
Bunuel wrote:

Tough and Tricky questions: Divisibility/Multiples/Factors.

How many factors does $$x$$ have, if $$x$$ is a positive integer?

(1) $$x = p^n$$, where $$p$$ is a prime number

(2) $$n^n = n + n$$, where $$n$$ is a positive integer

Kudos for a correct solution.

Official Solution:

How many factors does $$x$$ have, if $$x$$ is a positive integer?

We cannot easily rephrase the question. Note that we may not need to know $$x$$ in order to know how many factors it has.

Statement (1): INSUFFICIENT. Without knowing the value of $$n$$, we cannot determine the number of factors $$x$$ has.

Statement (2): INSUFFICIENT. This statement by itself is unconnected to the question, because the statement involves only the variable $$n$$, whereas the question only involves the variable $$x$$.

Statements (1) and (2) TOGETHER: SUFFICIENT. First, we should analyze the second statement further, to see whether we can find a unique value of $$n$$.

Since $$n$$ is a positive integer, we can test simple positive integers in an organized fashion, checking for equality of the two sides of the equation.

$$1^1 = 1 + 1$$? No.

$$2^2 = 2 + 2$$? Yes.

$$3^3 = 3 + 3$$? No.

$$4^4 = 4 + 4$$? No.

Notice that the left side of the equation is growing at a much faster rate than the right side, so the equation will not be true for any higher possible values of $$n$$. Thus, we can determine that the value of $$n$$ is 2.

Now, we do not know the value of $$p$$, nor of $$x$$, but we do now know that $$x = p^2$$, with $$p$$ as a prime number. Since a prime number has no factors other than 1 and itself, we can see that $$x$$ has no factors other than 1, $$p$$, and $$p^2$$. Thus, $$x$$ has exactly 3 factors, and we can answer the question definitively.

@Bunel, if we know P is prime, and all prime numbers have only three factors, and exponents do not produce new factors, why cant we say that # factors for N = 3 since N itself the just a prime number to a power?
Intern
Joined: 17 Apr 2016
Posts: 1
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

31 May 2016, 18:39
akumar5 wrote:
x is a positive integer

from statement 1 : x=p^n, where p is prime
so the no of factors will be n+1.
value of n will lead to complete solution.

from statement 2: n^n=n+n, where n is positive integer
n^n=2n
n^n-2n=0
n(((n^(n-1))-2)=0
hence either n=0 or n^(n-1)=2 that is n=2

combining both statements will yield two different no of factors for two different value of n.

It said n is a positive integer, so n=|=0
Ans-C

Sent from my XT1033 using GMAT Club Forum mobile app
Current Student
Joined: 12 Aug 2015
Posts: 2638
Schools: Boston U '20 (M)
GRE 1: Q169 V154
How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

24 Aug 2016, 06:55
Such an amazing Question
Here is y approach =>
Here we need to get the number of factors of x for x >0 and it is an integer
Statement 1 -> x=prime ^n okay if n=2=> factors =3 ;
n=4 => factors =5
Rule => if X= A^a*B^b where A and B are PRIME=> Number of factors of = (a+1)*(b+1)
Hence Insuff
Statement 2 => here only value possible is n=2
NOTE => 2 is a funny number =>
Its the only even prime
only number for which the square is twice the number
Its one of the two numbers for which number of divisors = number itself (other being 1)
2,3 are the only two consecutive numbers that are prime
and much more...

here though we have no clue on x => insuff
Combining them => x=Prime ^2 => number of factors =(2+1)=> 3
Smash that C
_________________

MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE!

Getting into HOLLYWOOD with an MBA!

The MOST AFFORDABLE MBA programs!

STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)

AVERAGE GRE Scores At The Top Business Schools!

Non-Human User
Joined: 09 Sep 2013
Posts: 8542
Re: How many factors does x have, if x is a positive integer ?  [#permalink]

### Show Tags

03 Jan 2018, 00:32
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: How many factors does x have, if x is a positive integer ? &nbs [#permalink] 03 Jan 2018, 00:32
Display posts from previous: Sort by