Find the GCD of the elements of the set $n^{7}-n$ with $\mathrm{n}$ an integer.
Let $d$ be the GCD.

\[
\begin{gathered}
2^{7}-2=126=2 * 3 * 3 * 7 \\
3^{7}-3=2184=3 * 2 * 2 * 2 * 7 * 13
\end{gathered}
\]

Thus $d$ divides $2^{*} 3^{*} 7=42$.

Let's show that in fact $d=42$.

Modulo 2: By Fermat's Little Theorem, $n^{7}=n^{5}=n^{3}=n[2]$

Modulo $3: n^{7}=n^{4}=n[3]$

Modulo $7: n^{7}=n[7]$

Therefore, $42 \mid d$ and thus $d=42$.

## Tool: Consecutive Powers

Theorem 18. For any natural number $a$, there is no perfect square between $a^{2}$ and $(a+1)^{2}$.

Theorem 19. For any natural number $a$, there is no $k$-th power between $a^{k}$ and $(a+1)^{k}$.