Calculating 7^7^7^7^7^7^7 mod 100 What is $$\large 7^{7^{7^{7^{7^{7^7}}}}} \pmod{100}$$ I'm not much of a number theorist and I saw this mentioned on the internet somewhere. Should be doable by hand.

$7^4 = 2401 \equiv 1 \pmod{100}$, so you only need to calculate $7^{7^{7^{7^{7^7}}}} \pmod{4}$. We know that $7 \equiv -1 \pmod 4$ and $7^{7^{7^{7^7}}}$ is odd, so $7^{7^{7^{7^{7^7}}}} \equiv -1 \equiv 3 \pmod 4$, and then $$7^{7^{7^{7^{7^{7^7}}}}} \equiv 7^3 \equiv 43 \pmod {100}$$