Prove that one of every three consecutive integers is divisible by 3.

Ans:  
n,n+1,n+2 be three consecutive positive integers We know that n  is of the form 3q, 3q +1, 3q + 2
So we have the following cases

Case – I  when n = 3q 

  In the this case, n is divisible by 3 but n + 1 and n + 2 are not divisible by 3

Case -  II When n = 3q + 1
Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not divisible by 3

     Case – III  When n = 3q +2
                   Sub n = 2 = 3q +1 +2 = 3(q +1) is divisible by 3. but n and n+1 are not                           divisible by 3

  Hence one of n, n + 1 and n + 2 is divisible by 3