[Math] Find The Remainder When $100^{100}$ Is Divided By 7

Question

Find the remainder when $100^{100}$ is divided by $7$.

I tried finding a pattern with the residues but it took a lot of time and I haven't found anything. Any answers?
Thanks!

Answer 1

\begin{align} 100 &=2 \pmod 7 \\ 2^1 &=2 \pmod 7 \\ 2^2 &=4 \pmod 7 \\ 2^3 &=1 \pmod 7 \end{align}

$$ 2^{100} = 2^{ 3*33+1 }={ 2 }^{ 99 }\cdot 2 = 2 \pmod 7$$