Prove that if n is any integer. Then either 3 |n or 3| ((n^2)-1)

Question

Case proof

Answer 1

There are 3 cases to consider:  n is a multiple of 3, n is one more that a multiple of 3 (n = 3k+1), or n is 2 more than a multiple of 3 (n = 3k+2).

 

Case 1:  n is a multiple of 3

 

             Then n = 3k, for some integer, k.  So, n is divisible by 3.  

 

Case 2:  n = 3k + 1, for some integer, k

 

             Then n² - 1 = (n+1)(n-1) = (3k+2)(3k) = 3(3k²+2k)

 

             So, n²-1 is divisible by 3

 

Case 3:  n = 3k+2, where k is an integer

 

             Then, n²-1 = (n+1)(n-1) = (3k+3)(3k+1) = 3(k+1)²

 

             So, n² -1 is divisible by 3.

 

Therefore, for any integer, n, n is divisible by 3 or n² - 1 is divisible by 3.  

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