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

Now, $n^2+n+1=2\cdot\dfrac{n(n+1)}2+1$ is odd

So, $4|(n^4-n)\iff4|n(n-1)$
Working in $\mathbb{Z}\_{4}$ we find $\bar{2}^{4}-\bar{2}=\bar{2}$
showing that $4\nmid\left(4m+2\right)^{4}-\left(4m+2\right)$ for
each integer $m$.