Equivalence of Definitions of Even Integer

$(1)$ $(2)$
By definition of divisor, $n$ is divisible by $2$ :
 * $n = 2 r$

where $r \in \Z$.

Thus definition 1 is logically equivalent to definition 2.

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

Setting $y = 0$:
 * $x \equiv 0 \pmod 2 \iff \exists r \in \Z: x = 2 r$

Thus definition 2 is logically equivalent to definition 3.