<s>[INST] Prove that for any integer, $n^2 + 5$ is not divisible by $4$. So I got that there is two cases: odd or even. 
If odd then say $n^2$ is $(2k+1)^2 = 4k^2 + 4k + 1.$ 
then $4k^2 + 4k + 1 + 5$ would need to be divisible by 4 and I don't know where to go from there. 
 [/INST] When $n$ is even,$n^2$ should also be even.Then $n^2+5$ should be odd,so it cannot be divided by 4.
When $n$ is odd,say $n=2k+1$,then $n^2+5=(2k+1)^2+5=4k^2+4k+6$.Since $4k^2$ and $4k$ can be divided by 4,this simply is about whether 6 can be divided by 4. 
</s>