<s>[INST] Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$ 
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$

I can not understand how to prove it. Please help me. 
 [/INST] $$\text{Here is a different way out; Recall that } \displaystyle \sum_{k=1}^m k = \dfrac{m(m+1)}2$$
$$\text{Hence, we have }1 + 2 + \cdots +(n-1) = \dfrac{n(n-1)}2 \text{ and } 1 + 2 + \cdots +(n+1)  = \dfrac{(n+1)(n+2)}2$$
$$\text{Putting the above two together, we get that}$$
$$\color{red}{\left(1 + 2 + \cdots +(n-1) \right) \left(1 + 2 + \cdots + n + (n+1)\right)} = \color{blue}{\dfrac{n(n^2-1)(n+2)}4}$$
$$\text{ Clearly, the }\color{red}{\text{left side is an integer}}\text{ and hence }\color{blue}{\text{right side is also an integer}}.$$
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