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.

First note that $$7^4 \equiv 1 \pmod{100}$$
Hence, we get that $$7^n \equiv \begin{cases}1 \pmod{100} & n \equiv 0 \pmod4\\ 7 \pmod{100} & n \equiv 1 \pmod4\\ 49 \pmod{100} & n \equiv 2 \pmod4\\ 43 \pmod{100} & n \equiv 3 \pmod4 \end{cases}$$
Hence, all we need to figure out is $7^{7^{100}} \pmod 4$. Since $7 \equiv (-1) \pmod4$, we have that $$7^{7^{100}} \equiv -1^{\text{odd}} \pmod 4 \equiv -1 \pmod 4 \equiv 3 \pmod 4$$
Hence, $$7^{7^{7^{100}}} \equiv 43 \pmod{100}$$
