$63^{63^{63}} \mod 100$ 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?

$(63)^7 = (60+3)^7 = (60^7 + {7\choose 1} 60^6\cdot 3 + \cdots + {7\choose 6} 60\cdot 3^6 + 3^7 $
All the terms to the left of the last 2 are equivalent to 0 modulo 100, and can be dropped.
Leaving $7\cdot 60\cdot 3^6 + 3^7\pmod {100}$
We only need to worry about the one figure for the first term since we know it will end in 0.
$3^4 = 81, 3^6\equiv 9 \pmod {10}$
$6\cdot 7 \cdot 9 \cdot 10 + 81\cdot 27\equiv 80 + 87 \equiv 67 \pmod {100}$
