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.
 A: 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.
A: Note that in this case
you have a direct proof:
$n^3-n
= n(n^2-1)
=n(n-1)(n+1)
$
and this is the product of $3$
consecutive integers,
so one of them must be divisible by $3$.
Though this is not what you asked,
I find it often useful
to know more than one proof
of a result,
because this gives me additional insights
which can be useful in other situations.
As is often true,
nothing about this proof is original.
A: $a \vert b$ means $a$ divides $b$, i.e., $b$ is a multiple of $a$, i.e., $b = ak$, where $k \in \mathbb{Z}$.
To prove by induction, prove for $n=1$ and note that
$$\left((n+1)^3 - (n+1) \right) - \left(n^3 - n\right) = \left(n^3 + 3n^2 + 3n + 1 - (n+1)\right) - n^3 + n = 3n^2 + 3n$$
Now use induction hypothesis to conclude what you want.
