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

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If n is a positive integer, is n^3 – n divisible by 4? [#permalink] New post 18 Aug 2009, 07:53
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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
[Reveal] Spoiler: OA
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Re: GMAT Official Guide 12th Edition DS Question 170 [#permalink] New post 05 Sep 2009, 18:35
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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.

Answer: A.
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Re: GMAT Official Guide 12th Edition DS Question 170 [#permalink] New post 06 Sep 2009, 03:14
[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
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Re: GMAT Official Guide 12th Edition DS Question 170 [#permalink] New post 06 Sep 2009, 10:11
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
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Re: GMAT Official Guide 12th Edition DS Question 170 [#permalink] New post 23 Nov 2011, 13:27
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.
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Re: GMAT Official Guide 12th Edition DS Question 170 [#permalink] New post 14 Jan 2012, 05:53
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] New post 02 Jun 2014, 23:54
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
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Re: If n is a positive integer, is n^3 – n divisible by 4? [#permalink] New post 03 Jun 2014, 00:43
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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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NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: If n is a positive integer, is n^3 – n divisible by 4?   [#permalink] 03 Jun 2014, 00:43
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