# show that any positive integer is of the form 3q ,or 3q +1 or, 3q+q for some integer q

Solution :     Let a be any positive integer and b=3 .  Applying division Lemma with a and b =3

We have

where   $\\ a= 3q+r \\ 0\leq r< 3$

$\\ \Rightarrow a=3q+0$ or ,     $a=3q+1$ or ,    $a=3q+2$

$\\ \Rightarrow a=3q$     or ,    $a=3q +1$   or ,$a=3q +2$ for some integer q

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