Integer

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dreamv: Senior | Next Rank: 100 Posts; Posts: 85; Joined: Thu Jan 12, 2012 7:18 pm; Thanked: 1 times

by dreamv » Sat Jan 14, 2012 10:11 pm

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

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

I was able to find only n=2, please help.

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Source: Beat The GMAT — Problem Solving | Sat Jan 14, 2012 10:11 pm

ArunangsuSahu: Master | Next Rank: 500 Posts; Posts: 382; Joined: Thu Mar 31, 2011 5:47 pm; Thanked: 15 times

by ArunangsuSahu » Sat Jan 14, 2012 10:13 pm

think about n=4,2

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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 » Sat Jan 14, 2012 10:15 pm

I posted an explanation here:

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Jan 14, 2012 10:28 pm

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

Note that, 2^n is a power of 2 only. Hence, any integer equal to 2^n must be a power of 2 only.

Now, n is an integer. Hence, for n^2 to be a power of 2 only n must be a power of 2. Say, n = 2^m.

Hence, 2^n = 2^(2^m) and n^2 = (2^m)^2 = 2^(2m)
Hence, 2^(2^m) = 2^(2m)
Or, 2^m = 2m

Only possible solutions for m are m = 1 and m = 2.
Hence, n = 2 and n = 4.

The correct answer is C.

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