$63^{63^{63}} \mod 100$ I need to find $63^{63^{63}} \bmod 100$. 
This is what I've got so far: 
Since $\gcd(63,100)=1$ we can use Euler's theorem. We have $\phi (100)=40$ so $63^{40} \equiv 1 \mod 100$ 
Again $\gcd(16,100)=1$ and $\phi (40)=16$, that is $63^{16} \equiv 1 \mod 40$
Using this I got that $63^{63} \equiv 7 \mod 40 $ which led me to $63^{63^{63}} \equiv 63^7 \mod 100$
I'm stuck here and don't know what to do next, what could I do now?

Use the fast exponentiation algorithm:
\begin{array}{rrrr}
n& x^{2^k} && P\bmod 100 \\\hline
7 & 63 && 63 \\
3 & 69 &&69\cdot 63\equiv 47 \\
1 & 61 && 61\cdot 47\equiv \color{red}{67} \\\hline
\end{array}
