Fermat's little theorem

Statement

Use Fermat’s little theorem to find the remainder of $2^{100}$ upon division by $7$.

Solution

Since $7$ is prime and does not divide $2$, Fermat gives $2^{6}\equiv 1\pmod 7$. Write $100=6\cdot 16+4$, so $2^{100}=\bigl(2^{6}\bigr)^{16}\cdot 2^{4}\equiv 1^{16}\cdot 16\equiv 16\equiv \boxed{2}\pmod 7.$