Prove by induction $\vphantom{\Large A}3\mid\left(n^{3} - n\right)$ So I'm just studying for my midterm and I came across this exercise:

Prove by mathematical induction that
  $\vphantom{\Large A}3\mid\left(n^{3} - n\right)$ for every positive integer $n$.

What does the pipe symbol "$\mid$" mean? I have never seen it before.

If $n = 1$, then $n^3 - n = 0 = 3 \times 0 $. So, result holds for the base case. Now let us suppose your statement holdsl for $n \in \mathbb{N}$. In other words, suppose
$$ n^3 - n = 3k_0 $$
We want to show $(n+1)^3 - (n+1) = 3k_1 $ for some $k_1 $.
But, Notice
$$ (n+1)^3 - (n+1) = n^3 + 3n + 3n^2 + 1 + n - 1 = n^3 - n + 3(n + n^2) + 3k_0 + 3(n + n^2) = 3(k_0 + n + n^2) \cong_3 0$$
Hence, the result follows by mathematical induction.
