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Intern  Joined: 28 Jul 2009
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If n is a positive integer, is n^3 – n divisible by 4?  [#permalink]

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3
9 00:00

Difficulty:   45% (medium)

Question Stats: 67% (02:06) correct 33% (02:21) wrong based on 369 sessions

### HideShow timer Statistics If n is a positive integer, is n^3 – n divisible by 4 ?

(1) n = 2k + 1, where k is an integer
(2) n^2 + n is divisible by 6
Math Expert V
Joined: 02 Sep 2009
Posts: 56300
Re: GMAT Official Guide 12th Edition DS Question 170  [#permalink]

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If n is a positive integer, is n^3 – n divisible by 4 ?

n^3-n=n(n^2-1)=(n-1)n(n+1), so we are asked whether the product of 3 consecutive integers is divisible bu 4.

(1) n = 2k + 1, where k is an integer --> n=odd --> as n is odd then both n-1 and n+1 are even hence (n-1)n(n+1) is divisible by 4. Sufficient.

(2) n^2 + n is divisible by 6 --> if n=2 then n^3-n=6 and the answer is NO but if n=3 then n^3-n=24 and the answer is YES. Not sufficient.

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Retired Moderator B
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Re: GMAT Official Guide 12th Edition DS Question 170  [#permalink]

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[quote="bambazoon"]This question is from the 12th ed. of the official guide and I think there's something wrong with the solution.
I'm sure others have done this question already, but I haven't been able to find a related topic.

The question goes:
If n is a positive integer, is n^3 - n divisible by 4?

1) n = 2k + 1, where k is an integer
2) n^2 + n is divisible by 6

(n-1)(n+1)n ........ 2 odd , even or 2 even,odd

from 1
suff

from 2

(n)(n+1) = 6n..........even , odd we dont know the 3rd..........insuff

A
Intern  Joined: 21 Aug 2009
Posts: 27
Re: GMAT Official Guide 12th Edition DS Question 170  [#permalink]

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Is n(n+1)(n-1) divisible by 4, n is positive
stat1: n= 2K+1 => n is odd, in that case (n-1) and (n+1) are even
=> the product should be divisible by 4 ....suff.

stat2:n(n+1) is divisible by 6 => one no. is even( multiple of 2) and other one is odd and odd no. is multiple of 3
=> the product may or may not be divisible by 4 (divisibility will hold only for even no.s those are multiple of 4)...insuff.
IMO A
GMAT Tutor G
Joined: 24 Jun 2008
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Re: OG, 11th edition, #147  [#permalink]

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1
1
luminousmocha wrote:
If n is a positive integer, is n^3-n divisible by 4?

1. n=2k+1, where k is an integer
2. n^2+n is divisible by 6

got stuck on this one for a while..

It's a divisibility question, and factoring (prime factorization or algebraic factorization, depending on the question) is pretty much always the key to divisibility questions. We have:

n^3 - n = n(n^2 - 1) = n(n-1)(n+1) = (n-1)(n)(n+1)

so we can see that n^3-n is just the product of the three consecutive integers, n-1, n and n+1.

Now S1 tells us that n is odd, so n-1 and n+1 are both even (divisible by 2), and therefore (n-1)(n)(n+1) is divisible by 4.

S2 is less useful; it tells us that n(n+1) is divisible by 2*3, but it might be that n=2, in which case the answer to our question is 'no', or that n=3, in which case the answer to our question is 'yes'. So S1 is sufficient, S2 is not.
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Manager  Joined: 05 Dec 2009
Posts: 101
Re: OG, 11th edition, #147  [#permalink]

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Agreed, should be A. I, incorrectly, thought that if n = -1 and so n(n-1)(n+1) = 0 is not divisible by 4. but 0 is divisible by any number.
Manager  Joined: 01 Feb 2010
Posts: 215
Re: OG, 11th edition, #147  [#permalink]

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Answer is A, if we put n=2k+1 in n(n+1)(n-1) then on simplifying it gives n(4k) which shows it is divisible by 4. Statement 2 is not useful to solve the problem.
Manager  Joined: 13 Dec 2009
Posts: 212
Re: OG, 11th edition, #147  [#permalink]

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1
stmt1: since the equation n^3 - n = n (n^2-1) = n(n-1)(n+1)
and replace it with n = 2k+1
(2k+1)2k(2k+2) = 4(2k+1)(k+1)
so it is clearly divisible by 4

stmt2 n^2+n = n(n+1) is divisible by 6 but still we have one more term n-1 in our equation which we dont know if even so insuff
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Intern  Joined: 23 Nov 2011
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Re: GMAT Official Guide 12th Edition DS Question 170  [#permalink]

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Hi there, I have a question: Can I not just test cases for statement 1? Say k= 1, 2 and 5, then n would be 3, 5 and 11, respectively. Then plug into n(cube)-n/4: 3 and 5 would work, 11 wouldn't --> insufficient.
Intern  Joined: 14 Sep 2010
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Re: GMAT Official Guide 12th Edition DS Question 170  [#permalink]

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1
If n is a positive integer, is n^3 - n divisible by 4?

1) n = 2k + 1, where k is an integer

2) n^2 + n is divisible by 6

Statement 1: n = 2k + 1

(2k + 1)^3 =

(4k^2 + 4k + 1)(2k + 1) =

8k^3 + (4k^2 + 8k^2) + (4k + 2k + 1).

n^3 - n is derived by subtracting
2k + 1 from the binomial.

8k^3 + (4k^2 + 8k^2) + (4k)

Each term of n^3 - n is divisible by 4 where n is odd.

Statement 2: n^2 + n is divisible by 6

If n = 2, n^2 + n = 6, and
(2)^3 - 2 = 6 is not divisible by 4.
However, if n = 3, n^2 + n = 12, and (3)^3 - 3 = 24 is divisible by 4.

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Re: If n is a positive integer, is n^3 – n divisible by 4?  [#permalink]

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first statement says k is an integer....so if k=0 then n will be 1 and the product of n(n-1)(n+1) will be zero....in that it is not divisible by 4

pls illustrate
Math Expert V
Joined: 02 Sep 2009
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Re: If n is a positive integer, is n^3 – n divisible by 4?  [#permalink]

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1
sayansarkar wrote:
first statement says k is an integer....so if k=0 then n will be 1 and the product of n(n-1)(n+1) will be zero....in that it is not divisible by 4

pls illustrate

Zero is divisible by EVERY integer except zero itself, (or, which is the same, zero is a multiple of every integer except zero itself).
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Re: If n is a positive integer, is n^3 – n divisible by 4?  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If n is a positive integer, is n^3 – n divisible by 4 ?

(1) n = 2k + 1, where k is an integer
(2) n^2 + n is divisible by 6

-> When you modify the condition and problem, n^3-n=4t?(t is a positive integer) --> (n-1)n(n+1)=4t? and n-1=even?. Since n-1=even and n+1=even, it becomes yes.
In 1), n-1=2k and it is sufficient.
In 2), from n(n+1)=6m(m is a positve integer), n=3 yes n=2 no, which is not sufficient. Therefore, the answer is A.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: If n is a positive integer, is n^3 – n divisible by 4?  [#permalink]

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_________________ Re: If n is a positive integer, is n^3 – n divisible by 4?   [#permalink] 26 Aug 2018, 23:45
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