Congruences doubt! 
What the rest of the division $2^{100}$ by $11$?

$$2^5=32\equiv10\equiv-1\pmod{11}\\(2^5)^{20}=2^{100}\equiv-1^{20}\;\text{or}\; (-1)^{20}$$??

We have
\begin{align*}
  2^{10} &= 2^5 \cdot 2^5 &\equiv (-1) \cdot (-1) &= (-1)^2  \pmod{11}\\
         &\vdots\\
  2^{100} &=2^5 \cdots 2^5 &\equiv (-1) \cdots (-1) &= (-1)^{20} \pmod{11}
\end{align*}
