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

 It is currently 23 May 2013, 00:03

# how many integers

Author Message
TAGS:
Manager
Status: GMAT Preperation
Joined: 04 Feb 2010
Posts: 105
Schools: HBS, Tuck
Followers: 1

Kudos [?]: 2 [0], given: 15

how many integers [#permalink]  29 Sep 2010, 06:13
00:00

Question Stats:

10% (00:00) correct 90% (00:33) wrong based on 0 sessions
for how many integers n is 2^n= n^2 ?
0
1
2
3
>3
[Reveal] Spoiler: OA
GMAT Club team member
Joined: 02 Sep 2009
Posts: 11566
Followers: 1797

Kudos [?]: 9578 [1] , given: 826

Re: how many integers [#permalink]  29 Sep 2010, 06:21
1
KUDOS
vanidhar wrote:
for how many integers n is 2^n= n^2 ?
0
1
2
3
>3

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 can not be more than 4. Also n can not be negative 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.
_________________
Retired Moderator
Joined: 02 Sep 2010
Posts: 815
Location: London
Followers: 56

Kudos [?]: 302 [0], given: 25

Re: how many integers [#permalink]  29 Sep 2010, 08:03
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

_________________
Senior Manager
Status: Time to step up the tempo
Joined: 24 Jun 2010
Posts: 420
Location: Milky way
Schools: ISB, Tepper - CMU, Chicago Booth, LSB
Followers: 6

Kudos [?]: 75 [0], given: 50

Re: how many integers [#permalink]  30 Sep 2010, 19:53
Easy one. +1 for option C. I helps to memorize the powers of 2 till 2^{10}.
_________________

Support GMAT Club by putting a GMAT Club badge on your blog

Re: how many integers   [#permalink] 30 Sep 2010, 19:53
Similar topics Replies Last post
Similar
Topics:
Set W= how many integers are there such that they are 3 17 Aug 2003, 10:39
How many integers are there between, but not including 2 13 Nov 2008, 03:01
PS-How many Positive integers? 5 28 Mar 2009, 23:09
1 How many integers? 16 04 Sep 2009, 06:15
1 How many integer pair 3 21 Oct 2009, 10:44
Display posts from previous: Sort by