Show that any positive odd integer is of the form

To Show: That any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q is any some integer.

Proof: Let a be any odd positive integer and b = 6.

Then, there exists integers q and r such that 

a = 6q + r, 0 ≤ r < 6 (by division algorithm)

⇒ a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4

But 6q or 6q + 2 or 6q + 4 are even positive integers

So, $a=6 q+1$ or $6 q+3$ or $6 q+5$

Hence it is proved that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5, where q is any some integer.