Proof by induction that certain number is an integer

Question

Prove that the number $\frac{2n^5}{5} + \frac{n^4}{2} - \frac{2n^3}{3} - \frac{7n}{30}$ is an integer $\forall n \in \mathbb{N}.$

Answer 1

The expression is

$$=2\cdot\frac{n^5-n}5-2\cdot\frac{n^3-n}3+\frac{n^4-n}2$$

But $n^4-n=n(n^3-1)=n(n-1)(n^2+n+1)=(n^2+n+1)(n^2-n)$

$$=2\cdot\frac{(n^5-n)}5-2\cdot\frac{(n^3-n)}3+(n^2+n+1)\cdot\frac{(n^2-n)}2$$

Now using induction or otherwise, we can prove prime $p|(n^p-n)$ for all integer $n$