Equivalence of Definitions of Even Integer

Theorem

The following definitions of the concept of Even Integer are equivalent:

Definition 1

An integer $n \in \Z$ is even if and only if it is divisible by $2$.

Definition 2

An integer $n \in \Z$ is even if and only if it is of the form:

$n = 2 r$

where $r \in \Z$ is an integer.

Definition 3

An integer $n \in \Z$ is even if and only if:

$x \equiv 0 \pmod 2$

where the notation denotes congruence modulo $2$.

Proof

$(1)$ if and only if $(2)$

By definition of divisor, $n$ is divisible by $2$ if and only if:

$n = 2 r$

where $r \in \Z$.

Thus definition 1 is logically equivalent to definition 2.

$\Box$

$(2)$ if and only if $(3)$

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.

$\blacksquare$