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

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Joined: 05 Feb 2011
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For how many integers n is 2^n = n^2 ? [#permalink]

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08 Apr 2011, 19:45
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For how many integers n is 2^n = n^2 ?

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-how-many-integers-n-is-2-n-n-101911.html
[Reveal] Spoiler: OA

Last edited by Bunuel on 20 Jul 2013, 11:33, edited 1 time in total.
Renamed the topic and edited the question.
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Re: For how many integers n is 2^n = n^2? [#permalink]

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08 Apr 2011, 20:14
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=> 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
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Re: For how many integers n is 2^n = n^2? [#permalink]

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08 Apr 2011, 22:19
There are only two integers that will satisfy this :

2 and 4, hence the answer is C.
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Re: For how many integers n is 2^n = n^2? [#permalink]

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21 Apr 2011, 19:42
n = 2 , 4 are only ones that satisfy given expression.

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Re: For how many integers n is 2^n = n^2? [#permalink]

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20 Jul 2013, 11:22
what about n=0, it does work and n is an integer .. I answered 3 then ..
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Re: For how many integers n is 2^n = n^2? [#permalink]

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20 Jul 2013, 11:35
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consomr123 wrote:
what about n=0, it does work and n is an integer .. I answered 3 then ..

$$2^0=1$$ and $$0^2=0$$ --> $$1\neq{0}$$.

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 $$8^2=4^3=2^6=64$$).

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-how-many-integers-n-is-2-n-n-101911.html
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Re: For how many integers n is 2^n = n^2?   [#permalink] 20 Jul 2013, 11:35
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