Odd Integer Modulo 4

Theorem
Let $n$ be an odd integer.

Then $n$ can be expressed either as:
 * $n = 4 k + 1$

or as:
 * $n = 4 k + 3$

Proof
By the Division Theorem, $n$ can be expressed as:
 * $n = 4 k + r$

where:
 * $k, r \in \Z$
 * $0 \le r < 4$

That is, one of the following holds:

Of these:

and:

and so are even by definition.

Then:

and:

and so are odd by definition.