Exponents and mod (Euler's theorem) I know how to compute $7^{402} \pmod{10}$ using Euler's theorem since $7$ and $10$ are relatively prime.
But is there an easy way without using a calculator to compute $12^{720} \pmod{10}$. I don't think Euler's Theorem can be applied because $12$ and $10$ are not relatively prime...
Also, for $5^{1806} \pmod{63}$, finding $\varphi(63)$ is kinda difficult. Is there an easy way to solve that?

First, note that $n\in\mathbb{N}\implies12^{n}\equiv2^{n}\pmod{10}$.
Second, note that $n\in\mathbb{N}\implies2^{n}\equiv2^{n+4}\pmod{10}$.
Hence $12^{720}\equiv2^{720}\equiv2^{716}\equiv\ldots\equiv2^{4}\equiv16\equiv6\pmod{10}$.
