Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 24 Aug 2016, 06:59

### 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 June
Open Detailed Calendar

# How many factors does the integer X have?

Author Message
TAGS:

### Hide Tags

Intern
Joined: 01 Sep 2010
Posts: 23
Followers: 1

Kudos [?]: 65 [1] , given: 8

How many factors does the integer X have? [#permalink]

### Show Tags

04 Oct 2010, 06:16
1
KUDOS
11
This post was
BOOKMARKED
00:00

Difficulty:

95% (hard)

Question Stats:

42% (02:58) correct 58% (02:12) wrong based on 488 sessions

### HideShow timer Statistics

How many factors does the integer x have?

(1) x^(x+3) = (2x)^(x-1)
(2) |3x-7| = 2x+2
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [1] , given: 10016

Re: DS: Factors of integer X [#permalink]

### Show Tags

04 Oct 2010, 06:30
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

(1) $$x^{(x+3)}=(2x)^{(x-1)}$$ --> $$x^{(x+3)}=2^{(x-1)}*x^{(x-1)}$$ --> $$x^{(x+3-x+1)}=2^{(x-1)}$$ --> $$x^4=2^{(x-1)}$$ --> $$x=1$$. Sufficient.

(2) $$|3x-7| = 2x+2$$ --> $$x=1$$ or $$x=9$$. Not sufficient.

_________________
Retired Moderator
Joined: 02 Sep 2010
Posts: 805
Location: London
Followers: 100

Kudos [?]: 853 [0], given: 25

Re: DS: Factors of integer X [#permalink]

### Show Tags

04 Oct 2010, 09:12
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

(1) $$x^{x+3}=(2x)^{x-1}$$
$$x^x * x^3 = 2^{x-1} * x^{-1} * x^x$$
Assuming x is not 0, we can cancel x^x out
$$x^4 = 2^{x-1}$$
The only solution to this is x=1.
1 has only one factor
Sufficient

(2) x>=7/3 then 3x-7=2x+2, x=9
x<7/3 then 3x-7=-2x-2 , x=1
x can have 1 factor or 3 factors
Insufficient

_________________
Director
Joined: 05 Sep 2010
Posts: 854
Followers: 82

Kudos [?]: 225 [0], given: 61

Re: How many factors does the integer X have? 1. X^(x+3) = [#permalink]

### Show Tags

14 Aug 2012, 00:47
how can the answer be A wont x = 0 satify equation no 1 ?...expert plz reply !!
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [2] , given: 10016

Re: How many factors does the integer X have? 1. X^(x+3) = [#permalink]

### Show Tags

14 Aug 2012, 01:24
2
KUDOS
Expert's post
how can the answer be A wont x = 0 satify equation no 1 ?...expert plz reply !!

If $$x=0$$ in (1) then the left hand side of the equation becomes $$0^{-1}=\frac{1}{0}=undefined$$, (while the right hand side becomes $$0^3=0$$). Remember you cannot raise zero to a negative power.

Hope it helps.
_________________
Manager
Joined: 03 Jul 2012
Posts: 139
GMAT 1: 710 Q50 V36
GPA: 3.9
WE: Programming (Computer Software)
Followers: 4

Kudos [?]: 78 [0], given: 16

Re: How many factors does the integer X have? 1. X^(x+3) = [#permalink]

### Show Tags

15 Aug 2012, 08:44
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

here, wat is meant by "How many factors" ?
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [0], given: 10016

Re: How many factors does the integer X have? 1. X^(x+3) = [#permalink]

### Show Tags

15 Aug 2012, 08:47
mehulsayani wrote:
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

here, wat is meant by "How many factors" ?

The question asks about (positive) divisors of some integer. For example 6 has 4 divisors (factors): 1, 2, 3, and 6.

For more check Number Theory chapter of Math Book: math-number-theory-88376.html

Hope it helps.
_________________
Manager
Status: Retaking next month
Affiliations: None
Joined: 05 Mar 2011
Posts: 229
Location: India
Concentration: Marketing, Entrepreneurship
GMAT 1: 570 Q42 V27
GPA: 3.01
WE: Sales (Manufacturing)
Followers: 5

Kudos [?]: 85 [0], given: 42

Re: DS: Factors of integer X [#permalink]

### Show Tags

17 Aug 2012, 05:58
Bunuel wrote:
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

(1) $$x^{(x+3)}=(2x)^{(x-1)}$$ --> $$x^{(x+3)}=2^{(x-1)}*x^{(x-1)}$$ --> $$x^{(x+3-x+1)}=2^{(x-1)}$$ --> $$x^4=2^{(x-1)}$$ --> $$x=1$$. Sufficient.

(2) $$|3x-7| = 2x+2$$ --> $$x=1$$ or $$x=9$$. Not sufficient.

$$x^4=2^{(x-1)}$$ --> $$x=1$$---------How do we arrive at this. How to solve these????? just by testing values vaguely or is there any particular mathematics we can do with this?????
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [0], given: 10016

Re: DS: Factors of integer X [#permalink]

### Show Tags

17 Aug 2012, 06:26
GMATPASSION wrote:
Bunuel wrote:
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

(1) $$x^{(x+3)}=(2x)^{(x-1)}$$ --> $$x^{(x+3)}=2^{(x-1)}*x^{(x-1)}$$ --> $$x^{(x+3-x+1)}=2^{(x-1)}$$ --> $$x^4=2^{(x-1)}$$ --> $$x=1$$. Sufficient.

(2) $$|3x-7| = 2x+2$$ --> $$x=1$$ or $$x=9$$. Not sufficient.

$$x^4=2^{(x-1)}$$ --> $$x=1$$---------How do we arrive at this. How to solve these????? just by testing values vaguely or is there any particular mathematics we can do with this?????

Number plugging is the best way to solve $$x^4=2^{(x-1)}$$:

x=0 does not satisfy the equation (check this: how-many-factors-does-the-integer-x-have-1-x-x-102180.html#p1112836).
x=1 satisfies the equation.

Now, if x>1 (2, 3, ...), then the left hand side of the equation is always greater than the right hand side of the equation: 2^4=16>2^1=2, 3^4=81>2^2=4, ... (as you can see LHS increases "faster" than RHS).

Hence, x=1 is the only positive integer solution of $$x^4=2^{(x-1)}$$.

Hope it helps.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [0], given: 10016

Re: How many factors does the integer X have? [#permalink]

### Show Tags

10 Jul 2013, 02:09
Bumping for review and further discussion.
_________________
Verbal Forum Moderator
Joined: 10 Oct 2012
Posts: 630
Followers: 76

Kudos [?]: 1005 [1] , given: 136

Re: DS: Factors of integer X [#permalink]

### Show Tags

10 Jul 2013, 04:36
1
KUDOS
GMATPASSION wrote:
Bunuel wrote:
How many factors does the integer X have?

1. X^(x+3) = (2x)^(x-1)
2. |3x-7| = 2x+2

(1) $$x^{(x+3)}=(2x)^{(x-1)}$$ --> $$x^{(x+3)}=2^{(x-1)}*x^{(x-1)}$$ --> $$x^{(x+3-x+1)}=2^{(x-1)}$$ --> $$x^4=2^{(x-1)}$$ --> $$x=1$$.

$$x^4=2^{(x-1)}$$ --> $$x=1$$---------How do we arrive at this. How to solve these????? just by testing values vaguely or is there any particular mathematics we can do with this?????

I believe Bunuel has already explained this. Just another way to look at it is this:

We end up with: $$x^4=2^{(x-1)} \to x = 2^{\frac{(x-1)}{4}}$$ .

We know that if x is an integer, the power of 2,i.e. $$\frac{x-1}{4}$$ has to be a non-negative integer.

I.$$\frac{x-1}{4}>0$$ As x equals some positive power of 2, x is even.Also, the only way x can be an integer is if the exponent of 2 is raised to an integer$$\to$$ x is a multiple of 4.

If (x-1) is a multiple of 4, then x has to be odd and this contradicts out previous statement, that x is even.

II. $$\frac{x-1}{4}=0$$$$\to x=1$$ and this satisfies the equality.
_________________
Senior Manager
Joined: 13 May 2013
Posts: 472
Followers: 3

Kudos [?]: 139 [0], given: 134

Re: How many factors does the integer X have? [#permalink]

### Show Tags

10 Jul 2013, 12:37
How many factors does the integer x have?

(1) x^(x+3) = (2x)^(x-1)
x^(x+3) = (2x)^(x-1)
x^(x+3) = x^(x-1) 2^(x-1)
(x^(x-1)) / x^(x+3) = x^(x-1) 2^(x-1) / (x^(x-1))
x^(x+3)-(x-1) = 2^(x-1)
x^(4) = 2^(x-1)
(This cannot be simplified any more. Thus, look for any value of x that will satisfy the equation. x=1 is the only value that will make x^(4) = 2^(x-1) valid)
SUFFICIENT

(2) |3x-7| = 2x+2

x≥7/3, x<7/3

x≥7/3
|3x-7| = 2x+2
(3x-7) = 2x+2
x=9 Valid

x<7/3
|3x-7| = 2x+2
-(3x-7) = 2x+2
-3x+7 = 2x+2
5=5x
x=1 Valid
We have two valid x values.
INSUFFICIENT

(A)

I'm not entirely sure why b) is insufficient. The stem doesn't tell us how many solutions we are looking for, it just asks for the number of factors. Isn't saying "number 2 has two factors" just as valid as saying "number one has one factor"?
Math Expert
Joined: 02 Sep 2009
Posts: 34416
Followers: 6248

Kudos [?]: 79372 [2] , given: 10016

Re: How many factors does the integer X have? [#permalink]

### Show Tags

10 Jul 2013, 12:41
2
KUDOS
Expert's post
WholeLottaLove wrote:
How many factors does the integer x have?

(1) x^(x+3) = (2x)^(x-1)
x^(x+3) = (2x)^(x-1)
x^(x+3) = x^(x-1) 2^(x-1)
(x^(x-1)) / x^(x+3) = x^(x-1) 2^(x-1) / (x^(x-1))
x^(x+3)-(x-1) = 2^(x-1)
x^(4) = 2^(x-1)
(This cannot be simplified any more. Thus, look for any value of x that will satisfy the equation. x=1 is the only value that will make x^(4) = 2^(x-1) valid)
SUFFICIENT

(2) |3x-7| = 2x+2

x≥7/3, x<7/3

x≥7/3
|3x-7| = 2x+2
(3x-7) = 2x+2
x=9 Valid

x<7/3
|3x-7| = 2x+2
-(3x-7) = 2x+2
-3x+7 = 2x+2
5=5x
x=1 Valid
We have two valid x values.
INSUFFICIENT

(A)

I'm not entirely sure why b) is insufficient. The stem doesn't tell us how many solutions we are looking for, it just asks for the number of factors. Isn't saying "number 2 has two factors" just as valid as saying "number one has one factor"?

When a DS question asks about the value, then the statement is sufficient ONLY if you can get the single numerical value.

From (2) we have that x=1 or x=9. If x=1, then the answer is 1 (1 has 1 factor) but if 9, then the answer is 3 (9 has 3 factors), so we have TWO different answers to the question. Therefore this statement is NOT sufficient.

Hope it's clear.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 11036
Followers: 509

Kudos [?]: 133 [0], given: 0

Re: How many factors does the integer X have? [#permalink]

### Show Tags

30 Sep 2014, 12:10
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 Legend
Joined: 09 Sep 2013
Posts: 11036
Followers: 509

Kudos [?]: 133 [0], given: 0

Re: How many factors does the integer X have? [#permalink]

### Show Tags

14 Jul 2016, 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.
_________________
Re: How many factors does the integer X have?   [#permalink] 14 Jul 2016, 12:56
Similar topics Replies Last post
Similar
Topics:
5 How many positive factors does the positive integer x have? 8 26 Dec 2015, 03:12
3 How many factors does x have, if x is a positive integer ? 7 17 Nov 2014, 12:29
3 How many even different factors does the integer P have? 4 19 May 2012, 08:55
5 How many prime factors does positive integer n have? 4 29 Mar 2011, 12:30
7 How many different factors does the integer n have? 5 05 Jan 2010, 12:18
Display posts from previous: Sort by