Cycling remainders of powers

Look for the first power that leaves remainder $1$.

1. $2^3=8$, which leaves remainder $1$ on division by $7$.
2. $100=3\times 33+1$, so $2^{100}=(2^3{)}^{33}\times 2$.
3. Modulo $7$ this is $1^{33}\times 2$.
4. The remainder is $2$.

Answer: $2$