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

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

Since $(2,11)=1$, then by Fermat's Little Theorem $$2^{10}\equiv1\pmod{11}\\(2^{10})^{10}=2^{100}\equiv1\pmod {11}$$
