Question

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

Answer 1

Let $n, n+1$ and $n+2$ be three consecutive positive integers

Also, we know that a positive integer $n$ is of the form $3 q, 3 q$ +1 or $3 q+2$.

Case I: When $n=3 q$

Here $n$ is clearly divisible by 3 .

But $(n+1)$ and $(n+2)$ are not divisible by 3 .

[When $(n+1)$ is divided by 3 , remainder is 1 and when $(n+2)$ is divided by 3 , the remainder is 2$]$

Case II: When $n=3 q+1$

Here $n+2=3 q+3=3(q+1)$

Clearly, it is divisible by 3 .

But $n$ and $(n+1)$ are not divisible by 3 .

Case III: When $n=3 q+2$

Here, $n+1=3 q+3=3(q+1)$

clearly, $(n+1)$ is divisible by 3

But $n$ and $(n+2)$ are not divisible by 3

Hence, One of every three consecutive positive integers is divisible by 3 .