prove-consecutive-divisible-integers-positive

Question 3
Prove that one of any three consecutive positive integers must be divisible by 3.

Any three consecutive positive integers will be of the form;
n, (n + 1) and (n + 2), where n is any natural number i.e., n = 1, 2, 3, …..
Any number n, on dividing with 3 leaves 0, 1 and 2 as the remainders.
So, n = 3p or 3p+1 or 3p+2

Case 1 : n = 3p

\Rightarrow

3 divides n

Case 2 : n = 3p+1 , then n+2 = (3p+1)+2 = 3p+3 = 3(p+1)

\Rightarrow

3 divides n+2

Case 3 : n = 3p+2, then n+1 = (3p+2)+1 = 3p+3 =3(p+1)

\Rightarrow

3 divides n+1

Hence, one of any three consecutive positive integers must be divisible by 3.