Calculate $2^{5104} \bmod 10$ using mental arithmetic I am practising interview for Jane Street's trader internship and I found the following question.

Question: Calculate $2^{5104} \bmod  10$ using mental arithmetic. 

I know that $2^5 \bmod 10 \equiv 2 \bmod 10.$
So, 
\begin{align*}
2^{5104} & = (2^5)^{1020} 2^4 \\
& \equiv 2^{1020}2^4 \\
& = (2^5)^{204}2^4 \\
& \equiv(2^5)^{40}2^8 \\
& \equiv (2^5)^8 2^8 \\
& \equiv (2^5)^3 2 \\
& \equiv 6 \bmod 10.
\end{align*}
However, I find the calculations above very taxing if I use mental arithmetic. I believe there should be a faster way to answer the question but I am not able to find one. 

The Chinese Remainder Theorem says:
$$2^{5104}\equiv x \pmod{2\cdot 5} \iff \begin{cases}2^{5104}\equiv x\pmod 2 \\ 2^{5104}\equiv x\pmod 5\end{cases}$$
It follows that $x\equiv 0\pmod 2$ and $x\equiv 1\pmod 5$.
Consequently $x\equiv 6\pmod{10}$.
