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

Question

(1) n is divisible by 6
(2) n = 8m + 4, where m is an integer

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GMATinsight · Answer

n^3 - n = n*(n^2 - 1) = (n-1)*n*(n+1)  (i.e. Product of three consecutive Integers)

Question: Is (n-1)*n*(n+1) divisible by 8?

Hint  :  Product of three consecutive Integers is divisible by 8 when
1) Smallest of them (i.e. n-1 in this case) is Even i.e. when n is ODD OR
2) when n itself is divisible by 8

Statement 1 : n is divisible by 6

Since n my be 6 in which case (n-1)*n*(n+1) is  NOT divisible by 8  
OR n my be 24 (multiple of 8 and 6 both) in which case (n-1)*n*(n+1) is  divisible by 8  hence

NOT SUFFICIENT

Statement 2 : n = 8m + 4, where m is an integer

i.e. n = 4(2m+1), since 2m+1 is ODD for all Integer values of m therefore, 
4(2m+1) will Neither be divisible by 8 nor will ever be ODD hence
Answer to the question is definitely NO hence

SUFFICIENT

Answer: Option B

Rabab36 · Answer

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

GMATinsight · Answer

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.

SaquibHGMATWhiz · Answer

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

WavuaMwakidedi · Answer

Is the answer D or B?

Bunuel · Answer

Answer is B.

(1) is not sufficient. For instance, if n = 6, the answer is NO. However, if n = 24, the answer is YES.

Hope it helps.

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

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