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