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Is positive integer n – 1 a multiple of 3?

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Is positive integer n – 1 a multiple of 3? [#permalink]  04 Jun 2012, 10:51
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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
[Reveal] Spoiler: OA
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Re: Is positive integer n – 1 a multiple of 3? [#permalink]  04 Jun 2012, 11:15
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Expert's post
ShreeCS wrote:
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

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.

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Re: Is positive integer n – 1 a multiple of 3? [#permalink]  12 Sep 2012, 05:17
1
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Bunuel wrote:
ShreeCS wrote:
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

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.

Hi Bunuel,
I got a question. "Is positive integer n-1 a multiple of 3" doesn't require a specific answer?
Through the Statement 2 we figure out that n-1 is a multiple of 3 only if n+1 would be as well, and the answer is yes, conversely if n would be a multiple of 3, in this case the answer is no.
Could me please explain better this doubt
Thank you
Math Expert
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Posts: 29802
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Kudos [?]: 53726 [1] , given: 8167

Re: Is positive integer n – 1 a multiple of 3? [#permalink]  12 Sep 2012, 05:29
1
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Expert's post
mario1987 wrote:
Bunuel wrote:
ShreeCS wrote:
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

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.

Hi Bunuel,
I got a question. "Is positive integer n-1 a multiple of 3" doesn't require a specific answer?
Through the Statement 2 we figure out that n-1 is a multiple of 3 only if n+1 would be as well, and the answer is yes, conversely if n would be a multiple of 3, in this case the answer is no.
Could me please explain better this doubt
Thank you

In a Yes/No Data Sufficiency question, statement(s) is sufficient if the answer is “always yes” or “always no” while a statement(s) is insufficient if the answer is "sometimes yes" and "sometimes no".

The question asks whether n-1 is a multiple of 3, and from (2) we have a definite NO answer to this question, so this statement is sufficient.

Hope it's clear.
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Re: Is positive integer n – 1 a multiple of 3? [#permalink]  21 Aug 2013, 02:15
Expert's post
Bumping for review and further discussion.
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Re: Is positive integer n – 1 a multiple of 3? [#permalink]  04 Sep 2014, 02:56
(1): Pick numbers. If n=5 then 5³-5 = 120 = multiple of 3, but n-1 = 4 no multiple of 3. And 4³-4 = 60 = multiple of 3 and 4-1 = 3 which is a multiple of 3. Insufficient. This will stay IS so the answer will be B or E.

(2) This is a bit trickier. First, simplify the expression:
n³+2n²+n = n(n²+2n+1) = n(n+1)² --> Multiple of 3. For this to be a multiple of 3, EITHER n OR n+1 is a multiple of 3 (both is not possible since they are consecutive integers).
Now pick numbers again: if n=5, then n+1 = 6 = multiple of 3, which satisfies the equation.
If n=3, then n is a multiple of 3, which again satisfies the equation.
Note that in both cases n - 1 is NOT a multiple of 3, which answers the question is n-1 a multiple of 3?

Re: Is positive integer n – 1 a multiple of 3?   [#permalink] 04 Sep 2014, 02:56
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