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by Anurag@Gurome » Mon Apr 18, 2011 8:16 pm
alltimeacheiver 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
(1) n^3 - n = n(n² - 1) = n(n - 1)(n + 1), which are consecutive integers.
So, n(n - 1)(n + 1) is a multiple of 3.
For any 3 consecutive integers, one of the 3 will always be a multiple of 3. But we cannot confirm if n - 1 is a multiple of 3.
So, (1) is NOT SUFFICIENT.

(2) n^3 + 2n² + n = n(n² + 2n + 1) = n(n + 1)² is a multiple of 3.
So, either n or (n + 1) is a multiple of 3. And if n or n + 1 is a multiple of 3, then (n - 1) cannot be a multiple of 3, which is SUFFICIENT to answer the questions asked.

The correct answer is B.
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by alltimeacheiver » Mon Apr 18, 2011 8:28 pm
sir thanks for the reply. but still not clear..Lets take numbers like n=3 so n(n+1)(n+1)= 3(4)(4)= 16*3= 48, which is multiple of 3 but if the number is n = 4 then n(n+1)(n+1)= 3(5)(5)= 25*3= 75= multiple of 3

Is it sounds nice with an example sir? Pls clarify
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by Anurag@Gurome » Mon Apr 18, 2011 8:33 pm
alltimeacheiver wrote:sir thanks for the reply. but still not clear..Lets take numbers like n=3 so n(n+1)(n+1)= 3(4)(4)= 16*3= 48, which is multiple of 3 but if the number is n = 4 then n(n+1)(n+1)= 3(5)(5)= 25*3= 75= multiple of 3

Is it sounds nice with an example sir? Pls clarify
We have to find if n - 1 is a multiple of 3.

If n = 3, n(n+1)(n+1)= 3(4)(4)= 16*3 = 48. Here n - 1 = 3 - 1 = 2, not a multiple of 3.
Please note that we cannot take n = 4, as neither n nor n + 1 would be a multiple of 3, because if n = 4, then n(n+1)(n+1)= 4(5)(5)= 25*4 = 100, which is not a multiple of 3.
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by alltimeacheiver » Tue Apr 19, 2011 10:43 pm
Pls clarify one thing again that n has to be in case above multiple of 3 which makes n-1 not multiple of 3. Ans to the statement would no and thus be the ans. pls confim
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by MAAJ » Wed Apr 20, 2011 3:48 pm
Note that statement (1) and (2) are very similar:

Statement (1)
(n - 1) n (n + 1), which are consecutive integers. So, (n - 1) n (n + 1) is a multiple of 3.
For any 3 consecutive integers, one of the 3 will always be a multiple of 3.

Statement (2)
n(n + 1)² = n (n + 1)(n + 1) is a multiple of 3. So, either n or (n + 1) is a multiple of 3.

From (n - 1) n (n + 1) we know that either n or (n + 1) is a multiple of 3. So the only one left is (n - 1) which will never be a multiple of 3, because the multiple of 3 will always lie (according to this statement) between n and (n + 1), and thus never in (n - 1). For example:

(2)*(3)*(4) = 24 (multiple of 3)
(1)*(2)*(3) = 6 (multiple of 3)

Hope it helps!
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