7. Prove that for any integer $n$, we have
(i) $6 \mid n(n+1)(n+2)$;
(ii) $8 \mid n(n+1)(n+2)(n+3)$;
(iii) $24 \mid n(n+1)(n+2)(n+3)$;
(iv) if $2 \nmid n$, then $8 \mid n^{2}-1$ and $24 \mid n\left(n^{2}-1\right)$;
(v) if $2 \nmid n$ and $3 \nmid n$, then $24 \mid n^{2}+23$;
(vi) $6 \mid n^{3}-n$;
(vii) $30 \mid n^{5}-n$;
(viii) $42 \mid n^{7}-n$;
(ix) $\frac{1}{5} n^{5}+\frac{1}{3} n^{3}+\frac{7}{15} n$ is an integer.