How to compute $7^{7^{7^{100}}} \bmod 100$? How to compute $7^{7^{7^{100}}} \bmod 100$? Is $$7^{7^{7^{100}}} \equiv7^{7^{\left(7^{100} \bmod  100\right)}} \bmod 100?$$
Thank you very much.

$7^4\equiv1\pmod{100}$
Let $$y=7^{7^{7^{100}}}$$
So, $7y=7^{(7^{7^{100}}+1)}\equiv1\pmod{100}$ as $(7^{7^{100}}+1)$ is divisible by $(7+1)=8$
Let us use the property of the successive convergents of a continued fraction. 
Now, $\frac{100}7=14+\frac27=14+\frac1{\frac72}=14+\frac1{3+\frac12}$
So, the previous convergent of $\frac{100}7$ is $14+\frac13=\frac{43}3$
So, $100\cdot3-7\cdot43=-1\implies y\equiv43\pmod {100}$
