<s>[INST] Show that $n^4-6n^3+11n^2-6n$ is divisible by $4$ for every integer $n$. So I am trying to refresh my memory when it comes to modular arithmetic but I am having some difficulties. The problem as states in the title is:

Show that $n^{4}-6n^{3}+11n^{2}-6n$ is divisible by $4$ för every
  integer $n$.

So what I first did was reformulating the problem in terms of congruences, in the following way: 
$$4\vert (n^{4}-6n^{3}+11n^{2}-6n) \iff  4\vert (n^{4}+11n^{2}-(6n^{3}+6n))\\ \hspace{6cm} \iff  n^{4}+11n^{2}\equiv 6n^{3}+6n\pmod 4$$
Now, one initial question I have is if it is correct that the only cases I need to check is:
$$n^{4}+11n^{2}\equiv 0\equiv 6n^{3}+6n\pmod 4 \\ n^{4}+11n^{2}\equiv 1\equiv 6n^{3}+6n\pmod 4\\ n^{4}+11n^{2}\equiv 2\equiv 6n^{3}+6n\pmod 4\\n^{4}+11n^{2}\equiv 3\equiv 6n^{3}+6n\pmod 4,$$
since $0,1,2,3$ are the possible remainders when considering division by $4$?
If this is true, it still seams like a really tedious approach to solving the problem. So I am wondering if there is some other way of solving it?
Thanks in advance. 
 [/INST] Continuing your idea (which isn't quite what I would have done) you can reduce the coefficients.
$n^{4}-6n^{3}+11n^{2}-6n \equiv 0 \pmod 4 \iff$
$n^4 + 11n^2 \equiv 6n^3 + 6n \pmod 4\iff$
$n^4 - n^2 \equiv 2n^3 + 2n\pmod 4\iff$
$n^2(n-1)(n+1) \equiv 2n(n^2 + 1)\mod 4\iff$
The check if $n\equiv 0\pmod 4$ then LHS and RHS are both $\equiv 0\pmod 4$.
If $n\equiv 3\equiv -1 \pmod 4$ or $n \equiv 1\pmod 4$ then $n\equiv \pm 1\pmod 4$ and $n\mp 1 \equiv 0\pmod 4$ so LHS is $\equiv 0 \pmod 4$.  And $n^2 +1\equiv 2\pmod 4$ an $2n \equiv \pm 2$ so RHS is $\equiv \pm 4 \equiv 0 \pmod 4$.
And if $n \equiv 2 \pmod 4$ then $n^2 \equiv 4 \equiv 0\pmod 4$ so LHS $\equiv 0\pmod 4$. meanwhile $2n\equiv 4\equiv 0 \pmod 4$. So RHS $\equiv 0 \pmod 4$.
So in all cases: $n^2(n+1)(n-1) \equiv 2n(n^2+1)\equiv 0\pmod 4$.
......
But I wouldn't have done it that way.
Perhaps the must informative, but most needlessly calculating way would be to factor.
$n^4 -6n^2 + 11n^2 -6n = n(n^3 - 6n^2 + 11n -6)=$
$n(n-1)(n^2 -5n^2 + 6)=n(n-1)(n-2)(n-3)$
Note: $n-3, n-2,n-1, n$ are four consecutive numbers!  So ONE of them is divisible by $4$.  So the product must be divisible by $4$.
That is an important lesson!  And an important strategy to learn.
But ... factoring is hard.
.......
If I were to do it my way, I'd reduce
$n^4 -6n^3 +11n^2 -6n \equiv n^4 + 2n^3 -n^2+2n\pmod 4$.
For $n\equiv 0,1,2,-1\pmod n$ we have $n^2 \equiv0,1,0,1\pmod n$ and $n^3\equiv 0,1,0,-1\pmod n$ and $n^4 \equiv 0,1,0,1\equiv n^2$ we'd have
$2n \equiv 0,2,0,-2\equiv 0,2,0,2$ and $2n^3 \equiv 0,2,0,2\equiv 2n$
So we have $n^4 + 2n^3 - n^2 + 2n \equiv n^2 +2n - n^2+ 2n\equiv 4n\equiv 0*n\equiv 0\pmod 4$.
.....
Which hints at Bill Dubuque's parity argument.
If $n$ is even than $n^4, n^3, n^2, 6n$ are all divisible by $4$.
If $n$ is odd then $n\equiv \pm 1$ and $n^4-6n^3 + 11n^2 -6n\equiv 1\mp 6+11 \mp 6=12 \mp 12\equiv 0\mp 0 \equiv 0\pmod 4$
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