<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] $n$ is even or odd.
$n$ is even: $n = 2m \Rightarrow n^2 + 5 = 4m^2 + 5 = 4(m^2 + 1) + 1$, which isn't divisible by $4$.
$n$ is odd: $n = 2m+1 \Rightarrow n^2 + 5 = 4m^2 + 4m + 1 + 5 = 4(m^2 + m + 1) + 2$, which isn't divisible by $4$. 
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