Can we find a solution to $(n-3)^3+n^3=(n+3)^3$?

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Show that, if \(n\) is an integer such that \[\begin{equation} (n-3)^3+n^3=(n+3)^3,\label{eq:1} \end{equation}\]

then \(n\) is even and \(n^2\) is a factor of \(54\). Deduce that there is no integer \(n\) which satisfies the equation \(\eqref{eq:1}\).

Show that, if \(n\) is an integer such that \[\begin{equation} (n-6)^3+n^3=(n+6)^3,\label{eq:2} \end{equation}\]

then \(n\) is even. Deduce that there is no integer \(n\) which satisfies the equation \(\eqref{eq:2}\).