What is the ten's digit of $7^{7^{7^{7^7}}}$ What is the ten's digit of $\zeta=7^{7^{7^{7^7}}}$. I got this question while doing binomial theorem. I think that $7^4=2401$ and we only need $\zeta\pmod{100}$. All I could think of is already presented (thought it is nothing actually), would you help? 

Note that 
$$7^2=49,\ 7^3=343,\ 7^4=2401,\ 7^5=168\color{red}{07}.$$
So, all we need is to find $7^{7^{7^{7}}}$ in mod $4$.
Now
$$7^{7^{7^{7}}}\equiv (-1)^{7^{7^{7}}}\equiv -1\equiv 3\pmod 4.$$
Hence, the answer we want is the same as the ten's digit of $343$, i.e. $\color{red}{4}$.
