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

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

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

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

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New post 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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New post 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.

Answer: C.

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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