2.1: Problem 21: Prove that \(n^{12} - a^{12}\) is divisible by \(13\) if \(n\) and \(a\) are prime to \(13\).
Proof
Suppose that \(n\) and \(a\) are coprime to \(13\). We want to show that \(n^{12} - a^{12}\) is divisible by \(13\). This means that we want to show that
$$
\begin{align*}
n^{12} - a^{12} \equiv 0 \pmod{13}
\end{align*}
$$
Since \((n,13) = 1\), then by Fermat’s theorem
$$
\begin{align*}
n^{12} \equiv 1 \pmod{13}
\end{align*}
$$
Moreover, since \((p,13)=1\), then by Fermat’s theorem
$$
\begin{align*}
a^{12} \equiv 1 \pmod{13}
\end{align*}
$$
Subtracting both equation, we see that
$$
\begin{align*}
n^{12} - a^{12} \equiv 0 \pmod{13}
\end{align*}
$$
which is what we wanted to show. \(\ \blacksquare\)