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

 It is currently 08 Dec 2019, 00:55 ### 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.  # For how many integers n is 2^n = n^2 ?

Author Message
TAGS:

### Hide Tags

Manager  Status: GMAT Preperation
Joined: 04 Feb 2010
Posts: 84
Concentration: Social Entrepreneurship, Social Entrepreneurship
GPA: 3
WE: Consulting (Insurance)
For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

2
24 00:00

Difficulty:   45% (medium)

Question Stats: 65% (01:23) correct 35% (01:12) wrong based on 604 sessions

### HideShow timer Statistics

For how many integers n is 2^n = n^2 ?

A. None
B. One
C. Two
D. Three
E. More than Three
Math Expert V
Joined: 02 Sep 2009
Posts: 59588

### Show Tags

8
11
For how many integers n is 2^n = n^2 ?
A. None
B. One
C. Two
D. Three
E. More than Three

$$2^n= n^2$$ is true for 2 integers:
$$n=2$$ --> $$2^2=2^2=4$$;
$$n=4$$ --> $$4^2=2^4=16$$.

Well, $$2^2=2^2=4$$ is obvious choices, then after trial and error you'll get $$4^2=2^4=16$$ as well. But how do we know that there are no more such numbers? You can notice that when $$n$$ is more than 4 then $$2^n$$ is always more than $$n^2$$ so $$n$$ cannot be more than 4. $$n$$ cannot be negative either as in this case $$2^n$$ won't be an integer whereas $$n^2$$ will be.

NOTE: I think it's worth remembering that $$4^2=16=2^4$$, I've seen several GMAT questions on number properties using this (another useful property $$8^2=4^3=2^6=64$$).

Hope it helps.
_________________
##### General Discussion
Retired Moderator Joined: 02 Sep 2010
Posts: 717
Location: London

### Show Tags

vanidhar wrote:
for how many integers n is 2^n= n^2 ?
0
1
2
3
>3

It helps to know that the function 2^x is more expansive than x^2 for large positive x and converges quickly to 0 for negative x. So we know we only have to check small values of x. For positive x, it is easy to see this is true for x=2,4 and then the function 2^x explodes

For negative x, 2^-1 is less than -1^2 already so no negative integers can satisfy the equality _________________
Manager  Joined: 18 Jan 2011
Posts: 207
Re: For how many integers n is 2^n = n^2?  [#permalink]

### Show Tags

1
1
=> 2^n = n^2
Taking nth root on both sides
=> 2 = (n^2)^1/n
=> 2 = n ^ 2/n
Lets consider positive even multiples of 2 for n (since LHS = 2)
For n = 2
=> 2 = 2 ^ 2/2 - First value that satisfier

For n = 4
=> 2 = 4 ^ 2/4 - Second value that satisfier

For n = 8
=> 2 = 8 ^ 2/8 - Doesnt satisfy

For n = 16
=> 2 = 16 ^ 2/16 - Doesnt satisfy

Two values. Ans = C
_________________
Good Luck!!!

***Help and be helped!!!****
Manager  Joined: 22 Feb 2016
Posts: 82
Location: India
Concentration: Economics, Healthcare
GMAT 1: 690 Q42 V47 GMAT 2: 710 Q47 V39 GPA: 3.57
Re: For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

This question might very well baffle us under the exam stress . So what is the methodology.
We can assume that the number wont be very large as - the bigger the numbers will get the difference between the two algebraic expression will increase.
we realise 2^0 is not equal to 0^2.
The continue with 1, 2, 3 4, 5, 6, 7, 8 by then you will get n=2, 4 suits the criterion, others don't and any bigger number will go super off limit.

trust me it took my 59 sec to do it using this long method.
Board of Directors D
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4834
Location: India
GPA: 3.5
Re: For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

vanidhar wrote:
For how many integers n is 2^n = n^2 ?

A. None
B. One
C. Two
D. Three
E. More than Three

$$n^2 = 2^n$$

So, $$n$$ = $$2^\frac{n}{2}$$

Now, n must be Even and a multiple of 2

Plug in the even values only 2 and 4 remains...

$$2$$ = $$2^\frac{2}{2}$$

$$4$$ = $$2^\frac{4}{2}$$

Hence, answer will be (C) 2

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Intern  B
Joined: 15 Apr 2017
Posts: 35
Location: United States
GMAT 1: 700 Q48 V38 Re: For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

why is the answer 2 and not 3? shouldn't zero be counted too?
Math Expert V
Joined: 02 Sep 2009
Posts: 59588
Re: For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

spatel2 wrote:
why is the answer 2 and not 3? shouldn't zero be counted too?

If n = 0:

2^n = 2^0 = 1 (recall that any nonzero number to the power of 0, is 1).
n^2 = 0^2 = 0.
_________________
Manager  P
Joined: 20 Jun 2018
Posts: 83
Location: Japan
Concentration: Strategy, Finance
GMAT 1: 660 Q49 V31 WE: Analyst (Energy and Utilities)
Re: For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

Hello chetan2u

Is there another approach to solving this question? I do not want to rely on hit and trial method.
Math Expert V
Joined: 02 Aug 2009
Posts: 8287
For how many integers n is 2^n = n^2 ?  [#permalink]

### Show Tags

vanidhar wrote:
For how many integers n is 2^n = n^2 ?

A. None
B. One
C. Two
D. Three
E. More than Three

akash7gupta11

$$2^n = n^2$$
What all does this tell you
(a) n cannot be negative as LHS 2^n will become fraction, while n^2 will remain integer.
(b) Since we have LHS as power of 2, the RHS or n will also be in terms of 2

So, let $$n=2^x$$....
$$2^{2^x}=(2^x)^2=2^{2x}.......2^x=2x.....2^{x-1}=x$$
Now moment x>2, the equality fails, so x=1 and 2

C
_________________ For how many integers n is 2^n = n^2 ?   [#permalink] 25 Nov 2019, 19:56
Display posts from previous: Sort by

# For how many integers n is 2^n = n^2 ?  