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Is positive integer n 1 a multiple of 3? (1) n^3 n is a

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Is positive integer n 1 a multiple of 3? (1) n^3 n is a [#permalink]

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17 Feb 2008, 12:49
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43% (01:33) correct 57% (00:27) wrong based on 195 sessions

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Is positive integer n – 1 a multiple of 3?

(1) n^3 – n is a multiple of 3
(2) n^3 + 2n^2+ n is a multiple of 3

Open discussion of this question is here: is-positive-integer-n-1-a-multiple-of-3-1-n-3-n-is-a-101676.html

Topic is locked.
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Feb 2012, 05:08, edited 2 times in total.
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Re: DS: Is positive integer n – 1 a multiple of 3? [#permalink]

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17 Feb 2008, 13:31
B.

From 1) (n-1)*n*(n+1) = 3*k

This is true for always for every value of n. Hence one cannot establish if n-1 is a multiple of 3. INSUFFICIENT

From 2) n (n+1) (n+1) = 3k.

Take some samples:
2*3*3 is a multiple of 3 but n-1 = 1 is not a multiple of 3.
3*4*4 is a multiple of 3 but n-1 = 2 is not a multiple of 3.
5*6*6 is a multiple of 3 but n-1 = 4 is not a multiple of 3.
6*7*7 is a multiple of 3 but n-1 = 5 is not a multiple of 3.
17*18*18 is a multiple of 3 but n-1 = 16 is not a multiple of 3.

Hence this eqn. is true for any n -1 where n-1 is not a multiple of 3. SUFFICIENT

P.S: It took me about 4 mins to get this one...

OA ?
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Re: Is positive integer n 1 a multiple of 3? (1) n^3 n is a [#permalink]

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13 Feb 2012, 00:04
n( n+1) (n+1) is multiple of 3 means out of n or n+1 one has to be mutliple of 3 if so than n-1 cannot be multilple of 3 bcoz in three consecutive integers there is only one mutilple of 3

here no need to go for diff values of n and prove it ............
correct me if i am wrong thanks
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Re: Is positive integer n 1 a multiple of 3? (1) n^3 n is a [#permalink]

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13 Feb 2012, 05:06
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pbull78 wrote:
n( n+1) (n+1) is multiple of 3 means out of n or n+1 one has to be mutliple of 3 if so than n-1 cannot be multilple of 3 bcoz in three consecutive integers there is only one mutilple of 3

here no need to go for diff values of n and prove it ............
correct me if i am wrong thanks

That's correct. Complete solution:

Is positive integer n – 1 a multiple of 3?

(1) n^3 – n is a multiple of 3 --> $$n^3-n=n(n^2-1)=(n-1)n(n+1)=3q$$. Now, $$n-1$$, $$n$$, and $$n+1$$ are 3 consecutive integers and one of them must be multiple of 3, so no wonder that their product is a multiple of 3. However we don't know which one is a multiple of 3. Not sufficient.

(2) $$n^3 + 2n^2+ n$$ is a multiple of 3 --> $$n^3 + 2n^2+ n=n(n^2+2n+1)=n(n+1)^2=3p$$ --> so either $$n$$ or $$n+1$$ is a multiple of 3, as out of 3 consecutive integers $$n-1$$, $$n$$, and $$n+1$$ only one is a multiple of 3 then knowing that it's either $$n$$ or $$n+1$$ tells us that $$n-1$$ IS NOT multiple of 3. Sufficient.

Open discussion of this question is here: is-positive-integer-n-1-a-multiple-of-3-1-n-3-n-is-a-101676.html
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Re: Is positive integer n 1 a multiple of 3? (1) n^3 n is a   [#permalink] 13 Feb 2012, 05:06
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