Equivalence of Definitions of Odd Integer

$(1)$ $(2)$
From the Division Theorem, setting $b = 2$:


 * $\forall a \in \Z: \exists! q, r \in \Z: a = 2 q + r, 0 \le r < 2$

Thus, either:
 * $\exists q \in \Z: n = 2 q$

or:
 * $\exists q \in \Z: n = 2 q + 1$

When $n = 2 q$, $n$ is even by definition.

When $n$ is not even, $n = 2 q + 1$

Likewise, when $n = 2 q + 1$ it follows that $n$ is not even.

Hence both definitions of odd integer are equivalent.

$(2)$ $(3)$
By definition of congruence modulo $2$:
 * $x \equiv y \pmod 2 \iff \exists r \in \Z: x - y = 2 r$

Setting $y = 1$:
 * $x \equiv 1 \pmod 2 \iff \exists r \in \Z: x - y = 2 r$

from which:
 * $x \equiv 1 \pmod 2 \iff \exists r \in \Z: x = 2 r + 1$

Thus definition 2 is logically equivalent to definition 3.