Prove that for all $n\in \mathbb{N}$ we have that $6$ divides $n^3-n$

Question

I'd like to know if my proof of the following is valid and if there's a shorter proof:

Prove that for all $n\in \mathbb{N}$ we have that $6$ divides $n^3-n$

Proof: We proceed by induction...

$P(1)=(1)^3-(1)=0$ which $6$ divides

Suppose the proposition is true for $n=k$, that is, there exists $j\in \mathbb{N}$ s.t. $k^3-k=6j$.

We show that $P(k+1)=(k+1)^3-(k+1)=6i$ for some $i$: $$(k+1)^3-(k+1)=k^3+3k^2+2k$$ $$=(k^3-k)+3k^2+3k$$ $$=\color{red}{6j+3k(k+1)}$$

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Now we show that $6$ divides $3k(k+1)$. Either $3k(k+1)$ is even (case 1), or odd (case 2).

(case 1) There exists $l$ s.t. $k=2l\implies3k(k+1)=6l(2l+1)$

(case 2) There exists $m$ s.t. $k=2m+1\implies3k(k+1)=3(2m+1)(2m+1+1)=6(2m^2+3m+1)$

Now, (case 1) and (case 2) $\implies 6$ divides $3k(k+1)\implies 3k(k+1)=\color{blue}{6q}$ for some $q\in\mathbb{N}$

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Next, $$P(k+1)=\color{red}{6j+3k(k+1)}=6j+\color{blue}{6q}=6(i)$$ Therefore, by induction $6$ divides $n^3-n$ for all $n\in \mathbb{N}$

Answer 1

Your proof looks fine, but there's definitely a shorter proof: $$n^3-n=(n-1)(n)(n+1) $$ At least one of those three factors is divisible by $2$, and at least one is divisible by $3$, so their product is divisible by $6$.

Answer 2

I think you're making things more complicated than needed. Just rewrite the given expression as follows: $$k^3 - k = k \times (k^2 - 1) = (k - 1) \times k \times (k + 1).$$ It is trivial that from n consecuitive numbers there is exactly one number dividable to n. So from the three consecutive numbers k - 1, k, and k + 1 there is exactly one number dividable to 3. And there is exactly one even number in the set {k, k + 1}. So $(k - 1) \times k \times (k + 1)$ is dividable to 2 and 3. Since 2 and 3 are relatively prime, $k^3 - k$ will be dividable to $2 \times 3 = 6$.

Answer 3

Since $n^3 \equiv n \pmod{2}$, we know that $2$ divides $n^3 - n$.

Now use induction to show that $n^3 - n$ is also divisible by $3$:

Base case: Certainly $3$ divides $n^3 - n$ when $n = 0$.

Inductive step:
Now suppose $3$ divides $n^3 - n$.
Algebra allows us to write

$\quad (n+1)^3 - (n+1) = n^3 + 3n^2 + 2n = n^3 - n + 3n^2 + 3n$

But then

$\quad [(n+1)^3 - (n+1)] \pmod{3}\equiv$
$\quad\quad [n^3 - n] \pmod{3} + [3n^2 + 3n]\pmod{3}\equiv$
$\quad\quad 0 \pmod{3} + 0 \pmod{3}\equiv 0\pmod{3}$

We conclude that $3$ also divides $n^3 -n$.

Since both $2$ and $3$ are prime factors, the product $6$ must also divide $n^3 -n$.