Show that, if \(n\) is an integer such that
\[\begin{equation*}
(n-3)^3+n^3=(n+3)^3,\label{eq:1sug}\tag{$1$}
\end{equation*}\]
then \(n\) is even and \(n^2\) is a factor of \(54\). Deduce that there is no integer \(n\) which satisfies the equation \(\eqref{eq:1sug}\).
Show that, if \(n\) is an integer such that \[\begin{equation*} (n-6)^3+n^3=(n+6)^3,\label{eq:2sug}\tag{$2$} \end{equation*}\]then \(n\) is even. Deduce that there is no integer \(n\) which satisfies the equation \(\eqref{eq:2sug}\).
Can \(n\) be even? Be odd? Would multiplying out the brackets help?
