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> Prove with induction that $n(n+1)(n+2)$ is divisible by $6$ for all $n \ge 1$.
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Test $n = 1$:

$$1(1+1)(1+2) = 6 $$

Hypothesis:

$$(\exists k \in \mathbb{N})(n(n+1)(n+2) = 6k)$$

This hypothesis is equivalent to

$$(\exists k \in \mathbb{N})(n^3+3n^2+2n = 6k)$$

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Prove for $n+1$:

$$(n+1)(n+2)(n+3)$$

Expand it all and notice:

$$\color{blue}{n^3}+\color{blue}{3n^2}+3n^2+9n+\color{blue}{2n}+6$$

Replace with hypothesis:

$$\color{blue}{6k}+3n^2+9n+6$$

Maybe I can factorize that quadratic:

$$6k+(3n+6)(n+1)$$

If I could take $6$ out as a common factor for the whole expression, I would be done. But I am unsure on how to proceed here. How can I achieve that?