Show that any positive odd integer is of the form
Question:

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

Solution:

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 = 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

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