Modular Exponentiation $a^n \bmod10$ for $a=\{2,3,...,9\}$

Question

$a^n\bmod 10\;$ for $a=\{2, 3,..., 9\}$

For $a=3,7,9$ I can use Euler's theorem but what about the rest.

I can see the patterns but how can I use those patterns as a proof?

Note: I can't use the Chinese remainder theorem.

Answer 1

Welcome!

It can be done by a really obvious induction – or simply saying you work in the ring $\mathbf Z/10\mathbf Z$.

Also, you can shorten the determination of the patterns.

All these powers $\{ a^n\mid n\in\mathbf N\}$ consist in cycles $\Gamma(a)$ of length $\ell(a)$. For instance, $\;\Gamma(2)=\{2,4,8,6\}$, so $\ell(2)=4$.

Now, for any power $k$, we have $\;\ell(a^k)=\dfrac{\ell(a)}{\gcd\bigl(k,\ell(a)}$, so

• $\ell(8)=4$. Indeed $\;\Gamma(8)=\{8,4,2,6\}$;
• $\ell(4)=2$: $\;\Gamma(4)=\{4,6\}$;
• $\ell(6)=\ell(2^4)=1$: $\;\Gamma(6)=\{6,4\}$.