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garuhape: Senior | Next Rank: 100 Posts; Posts: 74; Joined: Fri Dec 17, 2010 12:37 pm; Location: Munich; Thanked: 3 times

by garuhape » Sat Mar 26, 2011 2:32 am

Hi there,

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

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Source: Beat The GMAT — Data Sufficiency | Sat Mar 26, 2011 2:32 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Sat Mar 26, 2011 3:41 am

garuhape wrote:Hi there,

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

Statement 1: n^3 - n is a multiple of 3.
n^3 - n
= n(n^2 - 1)
= n(n+1)(n-1)
The above expression represents 3 consecutive integers: n-1, n, n+1.
Among every 3 consecutive integers, exactly 1 will be a multiple of 3.
Thus, n-1 could be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
Insufficient.

Statement 2: n^3 + 2n^2+ n is a multiple of 3.
n^3 + 2n^2 + n
= n(n^2 + 2n + 1)
= n(n+1)(n+1)
The above expression represents 2 consecutive integers: n and n+1.
Either n or n+1 must be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
If n+1 is a multiple of 3, then n-1 is not a multiple of 3.
Since in each case n-1 is not a multiple of 3, sufficient.

The correct answer is B.

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Sat Mar 26, 2011 3:44 am

garuhape wrote:Hi there,

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(nÂ² - 1) = n( n - 1)(n + 1) is a multiple of 3.
If n = 7, then n - 1 is a multiple of 3.
If n = 6, then n - 1 is not a multiple of 3.
No unique answer.

So, (1) is NOT SUFFICIENT.

(2) n^3 + 2nÂ²+ n = n(nÂ² + 2n + 1) = n(n + 1)Â² is a multiple of 3, which implies either n or (n + 1) is a multiple of 3.
If we consider that n is a multiple of 3: Say n = 6, then n - 1 cannot be a multiple of 3.
If we consider that (n + 1) is a multiple of 3: Say n + 1 = 6, then n - 1 cannot be a multiple of 3.
So, from here we can certainly say that n - 1 can NEVER be a multiple of 3; answer to main question is "No".
So, (2) is SUFFICIENT.

The correct answer is B.

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garuhape: Senior | Next Rank: 100 Posts; Posts: 74; Joined: Fri Dec 17, 2010 12:37 pm; Location: Munich; Thanked: 3 times

by garuhape » Sat Mar 26, 2011 6:55 am

GMATGuruNY wrote:
garuhape wrote:Hi there,

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
Statement 1: n^3 - n is a multiple of 3.
n^3 - n
= n(n^2 - 1)
= n(n+1)(n-1)
The above expression represents 3 consecutive integers: n-1, n, n+1.
Among every 3 consecutive integers, exactly 1 will be a multiple of 3.
Thus, n-1 could be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
Insufficient.

Statement 2: n^3 + 2n^2+ n is a multiple of 3.
n^3 + 2n^2 + n
= n(n^2 + 2n + 1)
= n(n+1)(n+1)
The above expression represents 2 consecutive integers: n and n+1.
Either n or n+1 must be a multiple of 3.
If n is a multiple of 3, then n-1 is not a multiple of 3.
If n+1 is a multiple of 3, then n-1 is not a multiple of 3.
Since in each case n-1 is not a multiple of 3, sufficient.

The correct answer is B.

Thanks a lot. Not just for this help, but for all your explanations. I'll have my test on Monday and you definitely helped to improve my score significantly.

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