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# If n is a positive integer, is n^3 - n divisible by 24?

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Re: If n is a positive integer, is n^3 - n divisible by 24? [#permalink]
Rabab36 wrote:
GMATinsight
in statement 2

when m=1,2,-1,-2
n=12,20,-4,-12

therefore n(n-1)(n+1) may or may not divisible by 24

Like I explained n(n-1)(n+1) represents product of three consecutive integers and you are not considering three consecutive integers hence your incorrect example is misleading you.

Statement 2 is Sufficient.
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Re: If n is a positive integer, is n^3 - n divisible by 24? [#permalink]
Bunuel wrote:
If n is a positive integer, is n^3 - n divisible by 24?

(1) n is divisible by 6
(2) n = 8m + 4, where m is an integer
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Solution:
Pre Analysis:
• n is a positive integer
• We are asked if $$n^3-n=n(n^2-1)=n(n-1)(n+1)=(n-1)n(n+1)$$, which is the product of 3 consecutive integers, is divisible by 24 or not
• We all know that product of 3 consecutive positive integers is always divisible by 6
• Now, to be divisible by 24, it has to be divisible by 4 too
• This is possible when $$n-1$$ will be even. This will make $$n+1$$ also even

Statement 1: n is divisible by 6
• If n is divisible by 6, then it has to be even
• Which makes $$n-1$$ odd and we can answer out question
• Thus, statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2: n = 8m + 4, where m is an integer
• Accordign to this statement, $$n=8m+4=even+even=even$$
• Which makes $$n-1$$ odd and we can answer out question
• Thus, statement 2 alone is also sufficient

Hence the right answer is Option D
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Re: If n is a positive integer, is n^3 - n divisible by 24? [#permalink]
Is the answer D or B?
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Re: If n is a positive integer, is n^3 - n divisible by 24? [#permalink]