Is zero odd or even?

For that, we can try all the axioms formulated for even numbers. I’ll use only four in this case.

Note: In this question, for the sake of my laziness, I will often use $N_e$ for even, and $N_o$ for odd.

Test 1:

An even number is always divisible by $2$.

We know that if $x,y\in \mathbb{Z}$
and $\dfrac{x}{y} \in \mathbb{Z},$ then $y$ is a divisor of $x$ (formally $y|x$).

Yes, both $0,2 \in \mathbb{Z}$ and yes, $\dfrac{0}{2}$ is $0$ which is an integer. Passed this one with flying colors!

Test 2:

$N_e + N_e$ results in $N_e$

Let’s try an even number here, say $2$. If the answer results in an even number, then $0$ will pass this test. $\ \ \ \ \underbrace{2}_{\large{N_e}} + 0 = \underbrace{2}_{N_e} \ \ \ $, so zero has passed this one!

Test 3:

$N_e + N_o$ results in $N_o$

$0 + \underbrace{1}_{N_o} = \underbrace{1}_{N_o}$

Passed this test too!

Test 4:

If $n$ is an integer of parity $P$, then $n - 2$ will also be an integer of parity $P$.

We know that $2$ is even, so $2 - 2$ or $0$ is also even.