If n is a positive integer, which of the following MUST be divisible

Question

If n is a positive integer, which of the following MUST be divisible by 6?

I.  n^3 - n 
II.  (n + 1)^2 - (n - 1)^2 
III.  n^3 - 4n

A. I only
B. II only
C. I and III only
D. I, II and III
E. None of the above

paragw · Answer

Here's my solution

TheAdmissionHub · Answer

(I)   n^3-n = n(n^2-1) = n(n+1)(n-1) 
If  n = 1 , the result is 0 which is divisible by 6.
If  n >= 2 , the result is always divisible by 6.
 True

(II)  (n+1)^2-(n-1)^2= n^2+1+2n-n^2-1+2n=4n 
If  n=1 , the result is not divisible by 6.
 False

(III)  n^3-4n=n(n^2-4)=n(n+2)(n-2)  
 If   n=1 , the result is not divisible by 6. 
 False

Answer: A

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