Exercise 2.1.24 ($n^{12} - a^{12}$ is divisible by $13$ if $n$ and $a$ are prime to $13$)

Question

Prove that $n^{12} - a^{12}$ is divisible by $13$ if $n$ and $a$ are prime to $13$.

Richard Ganaye · Accepted Answer

\begin{proof}  If $n$ and $a$ are prime to $13$, then
$$13 \mid n^{12} - 1,\qquad 13 \mid a^{12} - 1,$$
 by Fermat's theorem.
 
 Therefore
 $$13 \mid (n^{12} - 1) - (a^{12} - 1) = n^{12} - a^{12}.$$

\end{proof}

Exercise 2.1.24 ($n^{12} - a^{12}$ is divisible by $13$ if $n$ and $a$ are prime to $13$)

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Exercise 2.1.24 ($n^{12} - a^{12}$ is divisible by $13$ if $n$ and $a$ are prime to $13$)

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