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

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pleasehelp
 Post subject: for how many integers n is 2^n = n^2?
 Post Posted: Thu Oct 02, 2008 10:41 pm 
the answer is two. i answered 0.

which two? :)
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crackgmat
 Post subject:
 Post Posted: Thu Oct 02, 2008 11:53 pm 
For n = 2 and n = 4

Thanks
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Akshay
 Post subject:
 Post Posted: Fri Oct 03, 2008 10:58 am 
2 values of n...

n=2 & 4...

Hope m right... :roll: :roll:
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RonPurewal
 Post subject:
 Post Posted: Fri Oct 24, 2008 3:54 am 
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ManhattanGMAT Staff


Posts: 7146
yes, 2 and 4.
hopefully, 2 jumped out at you pretty fast; if n = 2, then the two sides of the equation are 2^2 and 2^2. i'm pretty sure that those are going to be equal.
you'll discover n = 4 through raw experimentation; there isn't any better way, unfortunately.

if you want to be confident that these are the only solutions, you have to watch the behavior of 2^n and n^2 as you get further and further away from 4. the pattern you'll observe is that 2^n begins to grow much, much faster than does n^2, making it clear that the two expressions won't be equal for any larger values.
the equality is definitely impossible for negative integers, because 2^(negative integer) is a fraction, while (negative integer)^2 is not. therefore, you don't have to worry about negative integers.
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be.a.true.winner
 Post subject: Re:
 Post Posted: Sat Nov 12, 2011 2:41 am 
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Students


Posts: 2
RonPurewal wrote:
yes, 2 and 4.

the equality is definitely impossible for negative integers, because 2^(negative integer) is a fraction, while (negative integer)^2 is not. therefore, you don't have to worry about negative integers.


Just to confirm, I think if the question did not mention "integers", there would be 3 possible solutions, right Ron?
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RonPurewal
 Post subject: Re: Re:
 Post Posted: Wed Nov 23, 2011 6:27 am 
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ManhattanGMAT Staff


Posts: 7146
be.a.true.winner wrote:
RonPurewal wrote:
yes, 2 and 4.

the equality is definitely impossible for negative integers, because 2^(negative integer) is a fraction, while (negative integer)^2 is not. therefore, you don't have to worry about negative integers.


Just to confirm, I think if the question did not mention "integers", there would be 3 possible solutions, right Ron?


yeah, there would be another solution (some number between -1 and 0). but that is way, way beyond the scope of the gmat.
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