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# Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n

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Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  26 Apr 2012, 03:05
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Is n^2 + 2n an integer?

(1) n^3 + 3n is an integer.
(2) 2n is an integer.

Source:Nova
[Reveal] Spoiler: OA

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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  26 Apr 2012, 03:31
Is n^2 + 2n an integer?

(1) n^3 + 3n is an integer --> if n=1 then the answer is YES. But if n^3+3n=1 then n=irrational \ number\approx{0.32}, for which n^2 + 2n\neq{integer}. So, when n is an integer then the answer is YES but if n is some irrational number then the answer is NO. Not sufficient.

(2) 2n is an integer --> n is either an integer (answer YES) or \frac{integer}{2} (answer NO). Not sufficient.

(1)+(2) From above n can only be an integer, which makes n^2 + 2n equal to an integer. Sufficient.

P.S. Not a good question, not a good source.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  26 Apr 2012, 04:33
Thanks for the explanation.After solving 50-60 questions, i also felt that the NOVA book, is not that good as explantions for some of the answers are confusing, and further it demoralises me when, I fail to understand/solve the problem.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  18 May 2012, 03:54
It is clear that statement a and b alone are not sufficient, however when we combine 1 and 2..

=> n3 + 3n = I; n^3 + 2n +n = I; n^3+n = I <given that 2n is an integer> = n3 + n can be integer only if n is integer.. therefore C.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  22 May 2012, 21:53
Quote:
(1)+(2) From above can only be an integer, which makes equal to an integer. Sufficient.

how did you arrive at above statement

Quote:
n3 + n can be integer only if n is integer..

Why ? bunnel calc. n for n^3 + 3n =1 , so in a similar way(i dont know how he calculated, may be he can tell) n^3+n=integer can be calculated
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  22 May 2012, 23:43
vikram4689 wrote:
Quote:
(1)+(2) From above can only be an integer, which makes equal to an integer. Sufficient.

how did you arrive at above statement

Quote:
n3 + n can be integer only if n is integer..

Why ? bunnel calc. n for n^3 + 3n =1 , so in a similar way(i dont know how he calculated, may be he can tell) n^3+n=integer can be calculated

From (1) n is either an integer or an irrational number;
From (2) n is either an integer or an \frac{integer}{2};

For (1)+(2) n can only be an integer.

Hope it's clear.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  22 May 2012, 23:52
Quote:
an irrational number;

how did you find that solution of n^3+3n=1 is an irrational no.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  22 May 2012, 23:58
vikram4689 wrote:
Quote:
an irrational number;

how did you find that solution of n^3+3n=1 is an irrational no.

Solution of n^3+3n=1 is not an integer or a fraction (try to prove it) so it must be an irrational number.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 00:04
on same lines, can i say that n^3+n=integer is also irrational
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 00:10
vikram4689 wrote:
on same lines, can i say that n^3+n=integer is also irrational

If n is an integer then n^3+n is always an integer. But n^3+n=integer does not necessarily means that n is an integer.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 00:41
Bunuel wrote:
vikram4689 wrote:
on same lines, can i say that n^3+n=integer is also irrational

If n is an integer then n^3+n is always an integer. But n^3+n=integer does not necessarily means that n is an integer.

Agreed, i am referring to above poster who said n3 + n can be integer only if n is integer but i was thinking that n can be irrational as well. Now how do i arrive at a conclusion.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 00:45
vikram4689 wrote:
Bunuel wrote:
vikram4689 wrote:
on same lines, can i say that n^3+n=integer is also irrational

If n is an integer then n^3+n is always an integer. But n^3+n=integer does not necessarily means that n is an integer.

Agreed, i am referring to above poster who said n3 + n can be integer only if n is integer but i was thinking that n can be irrational as well. Now who do i arrive at a conclusion.

Again, (1) says: n^3 + 3n is an integer. n^3+n=integer can occur when n=integer (for example n^3+n=2 --> n=1) as well as when n=irrational number (for example n^3+n=3 --> n=irrational number).
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 00:48
i totally understand the solution you have provided. i was reading below solution given by some other poster and i think he is wrong when he mentioned n3 + n can be integer only if n is integer.. . Although he mentions answer correctly
Quote:
It is clear that statement a and b alone are not sufficient, however when we combine 1 and 2..

=> n3 + 3n = I; n^3 + 2n +n = I; n^3+n = I <given that 2n is an integer> = n3 + n can be integer only if n is integer.. therefore C.
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Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n [#permalink]  23 May 2012, 11:39
between c & e i picked e. also but now C makes sense. in any case the OA provided is just wrong.
Re: Is n^2 + 2n an integer? (1) n^3 + 3n is an integer. (2) 2n   [#permalink] 23 May 2012, 11:39
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