Question

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

Answer 1

Consider a number $n,q$ and $r$ are positive integers. When $n$ is divided by 3 the quotient is $q$ and remainder $r$. So, by Euclid;s division algorithm,

$n=3q+r(0\le r<3)$ or $r=0,1,2,3$

At	$n_1$	Divisible by 3	$n_2=n_1+1$	Divisible by 3	$n_3=n_1+1$	Divisible by 3
$r=0$	$3q+0=3q$	Yes	$3q+1$	No	$3q+2$	No
$r=1$	$3q+1$	No	$3q+2$	No	$3q+3$ $=3(q+1)$ $=3m$	Yes
$r=2$	$3q+2$	No	$3q+3$ $=3(q+1)$ $=3m$	Yes	$3q+4$ $=3q+3+1$ $=3(q+1)+1$ $=3m+1$	No

So, one of any three consecutive positive integers is divisible by 3.