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# If n is a positive integer. Is n^3-n divisible by 24? 1) 2n has twice

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If n is a positive integer. Is n^3-n divisible by 24? 1) 2n has twice  [#permalink]

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15 May 2017, 04:05
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Difficulty:

95% (hard)

Question Stats:

42% (02:26) correct 58% (02:18) wrong based on 74 sessions

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If n is a positive integer. Is n^3-n divisible by 24?

1) 2n has twice as many factors as n has
2) n has exactly 2 distinct positive factors

source: http://www.GMATinsight.com

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Re: If n is a positive integer. Is n^3-n divisible by 24? 1) 2n has twice  [#permalink]

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15 May 2017, 04:57
GMATinsight wrote:
If n is a positive integer. Is n^3-n divisible by 24?

1) 2n has twice as many factors as n has
2) n has exactly 2 distinct positive factors

source: http://www.GMATinsight.com

Hi,

$$n^3-n=n(n^2-1)=n(n-1)(n+1)$$
(n-1)*n*(n+1) means three CONSECUTIVE integers..
Points to consider here..
1) ATLEAST one number will be multiple of 2 and one number multiple of 3..
2) If n is even number, n-1 and n+1 will be odd... So $$n^3-n$$ is div by ATLEAST 2*3=6..
3) If n is Odd, one of n-1 or n+1 is multiple of 2 and other multiple of atleast 4.. So $$n^3-n$$ is div by ATLEAST 2*4*3=24..

Let's see the statements..

1) 2n has twice as many factors as n has
This is possible ONLY when n is odd, so point (3) tells us that n^3-n is div by 24
Sufficient

2) n has exactly 2 distinct positive factors
This means n is prime number.
If n is 2, n^3-n=2^3-2=6.. so ans is NO
If n is 3,5 or any other odd prime number, ans will be YES as per point (3) above.
Insufficient

A
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Re: If n is a positive integer. Is n^3-n divisible by 24? 1) 2n has twice  [#permalink]

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08 Sep 2018, 06:00
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Re: If n is a positive integer. Is n^3-n divisible by 24? 1) 2n has twice   [#permalink] 08 Sep 2018, 06:00
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