Question: I need a little bit of help with this question: 

Show that there are an infinite number of positive integers n such that $n^4 - n$ is not divisible by 4.

Obviously, this is true for all $n$ except for *multiples of* 4. 

I assume that I'm supposed the problem either by contradiction or induction - If I perform induction, I reach the following steps:

$$(n+1)^4 - (n+1) = n^4 + 4n^3 + 6n^2 + 4n + 1 - (n+1) = 4k $$

Which gives us a polynomial that is not divisible by 4 into an integer except when $n = 4$. Is this enough to prove it? Or is there a step I'm missing?

Answer: 
For all positive integers $k$, $$(4k-2)^4-(4k-2)=4(64k^4-128k^3+96k^2-33k+4)+2$$
is not divisible by $4$.
$n^4 - n$ is divisible by $4$ for $n \equiv 0$ or $n \equiv 1 \bmod 4$.

$n^4 - n$ leaves remainder $2$ when divided by $4$ for $n \equiv 2$ or $n \equiv 3 \bmod 4$.