Integer N

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fangtray: Master | Next Rank: 500 Posts; Posts: 273; Joined: Thu Sep 08, 2011 6:50 amFollowed by: 3 members; Thanked: 5 times

by fangtray » Wed Apr 04, 2012 5:49 am

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

a. none
b. one
c. two
d. three
e. more than 3

this seemed like such an easy question. i plugged in 1 found it was not one of the integers, plugged in -1 and it also was not one of them. Plugged in 0 and that did not work. Figured it was NONE

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Source: Beat The GMAT — Problem Solving | Wed Apr 04, 2012 5:49 am

neelgandham: Community Manager; Posts: 1060; Joined: Fri May 13, 2011 6:46 amFollowed by: 52 members; Location: Utrecht, The Netherlands; Thanked: 318 times

by neelgandham » Wed Apr 04, 2012 5:54 am

fangtray wrote:For how many integers n is 2^n = n^2
a. none
b. one
c. two
d. three
e. more than 3

2^2 = 2^2
2^4 = 4^2, So the answer should be [spoiler]C = 2[/spoiler]

Anil Gandham
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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Wed Apr 04, 2012 6:20 am

I posted a solution here:

https://www.beatthegmat.com/exponents-t80313.html

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gmatmath: Junior | Next Rank: 30 Posts; Posts: 15; Joined: Mon Mar 26, 2012 12:14 am; Thanked: 3 times

by gmatmath » Fri Apr 06, 2012 8:12 am

We can solve this problem by trial and error method.
We will plug in values for n from 1,2,3,.... and chk for which value of n, 2^n = n^2.

1)n=1
2 = 1 FALSE

2)n=2
2^2 = 4; n^2 = 4 TRUE

3) n =3
2^3 = 8; 3^2 = 9 FALSE

4) n=4
2^4 = 16; 4^2=16 TRUE

5)n=5
2^5=32; 5^2=25 FALSE

6) n=6
2^6=64; 6^2 =36 FALSE

From this we can say that, as n increases, 2^n >> n^2.

hence for 2 values of n, this will hold true. SO the answer is option C

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Birottam Dutta: Master | Next Rank: 500 Posts; Posts: 342; Joined: Wed Jul 08, 2009 8:50 amFollowed by: 19 members; Thanked: 214 times; GMAT Score:740

by Birottam Dutta » Fri Apr 06, 2012 10:19 am

In addition to the answers above, you may also draw a graph of the two functions 2^N and N^2 and see where they intersect. There will be two points which gives the answer to the question.

Hence, answer C.

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ronnie1985: Legendary Member; Posts: 626; Joined: Fri Dec 23, 2011 2:50 amFollowed by: 10 members; Location: Ahmedabad; Thanked: 31 times

by ronnie1985 » Fri Apr 06, 2012 10:50 am

2 and 4 satisfy the eqn

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