Why is $n^2+4$ never divisible by $3$? Can somebody please explain why $n^2+4$ is never divisible by $3$?  I know there is an example with $n^2+1$, however a $4$ can be broken down to $3+1$, and factor out a three, which would be divisible by $4$.

Consider cases. For example, what happens if $n=3k$, $n=3k+1$, and $n=3k+2$?
