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