Theorem, One of every three consecutive positive integer is divisible by three

Let n-1,n and n+1 be three consecutive positive integers.

from division-algorithm We know that n can be written in the form of 3q,3q+1 or, 3q+2................. where q is any poitive integer.

In this situation three different cases arises

Case I:-

 When n=3q
here 3꘡n ,  but n-1 and n+1 are not divisible by 3.

Case II:-

 When n=3q+1
In this case, n-1=3q   means n-1 is divisible by 3.

here 3꘡n-1,  but n and n+1 are not divisible by 3.

Case III:-

 When n=3q+2
In this case, n+1=3q+2+1=3(q+1)

here 3꘡n+1   but n-1 and n are not divisible by 3.

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