Even and Odd proof: Direct proof conflict with Counter example Question: If $n \in \mathbb{Z}$ and $n^5 − n$ is even, then $n$ is even.
My solution: By counter example, let $n = 3$, then $3^5 - 3 = 240 =$ even, but $3 =$ odd.
However, if I use direct proof on this, it will be:
Suppose $n \in \mathbb{Z}$ and $n^5− n$ is even, then $n^5 − n = 2a$, where $a \in \mathbb{Z}$. 
It follows that $n(n^4 - 1) = 2a$, which yields $$n = 2\Biggl(\frac{a}{n^4 - 1}\Biggr)$$
Since both $a,n \in \mathbb{Z}$, the term inside the parentheses is also an integer. Hence, $n$ is even because it's a multiple of $2$.
It seems there is something wrong with my logic in the direct proof, because of the counter example...there must be something wrong here. Is it because that I can't have the integer $n$ on the right side to show this proof? But we are given that $n \in \mathbb{Z}$.
I will appreciate it very much if someone can help out.
Thanks

In fact, your own counterexample shows why the proof can't work (and it shouldn't work, because it's false).  In your example, you would have
$$
3^5-3 = 3(3^4-1) = 2(120)
$$
and so 
$$
3 = 2 \left(\frac{120}{3^4-1}\right).
$$
But $80$ doesn't divide $120$.
