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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]  29 Sep 2010, 05:13
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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
[Reveal] Spoiler: OA
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Re: how many integers [#permalink]  29 Sep 2010, 05:21
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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

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

Hope it helps.
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Re: how many integers [#permalink]  29 Sep 2010, 07:03
vanidhar wrote:
for how many integers n is 2^n= n^2 ?
0
1
2
3
>3

It helps to know that the function 2^x is more expansive than x^2 for large positive x and converges quickly to 0 for negative x.

So we know we only have to check small values of x. For positive x, it is easy to see this is true for x=2,4 and then the function 2^x explodes

For negative x, 2^-1 is less than -1^2 already so no negative integers can satisfy the equality

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Re: how many integers [#permalink]  30 Sep 2010, 18:53
Easy one. +1 for option C. I helps to memorize the powers of 2 till $$2^{10}$$.
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Re: For how many integers n is 2^n = n^2 ? [#permalink]  06 Oct 2014, 18:09
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Re: For how many integers n is 2^n = n^2 ? [#permalink]  07 Oct 2014, 02:22
$$2^2 = 2^2$$

$$4^2 = 2^4$$

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Re: For how many integers n is 2^n = n^2 ?   [#permalink] 07 Oct 2014, 02:22
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