Two questions about divisible I have two questions.
1) Why the relation $a^n\equiv a^{n+4k}\pmod{10}$ is true?
2) Let $(a,90)=1$. Which number can be $ x$ in $ x\mid a^4-1$? Answer is $240$. Why?

\begin{align}
3^0 & \equiv 1 \pmod{10} \\
3^1 & \equiv 3 \pmod{10} \\
3^2 & \equiv 9 \pmod{10} \\
3^3 & = 27 \equiv 7 \pmod{10} \\
3^4 & = 81 \equiv 1 \pmod{10}
\end{align}
We've returned to $1$ when the exponent is $4$.
Try this with other numbers is $\{1,3,7,9\}$ in place of $3$.  In each case we get the $4$th power congruent to $1$.  The four numbers $1,3,7,9$ are those that are coprime to $10$.
