Finding the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$ 
Find the remainder when $2^{100}+3^{100}+4^{100}+5^{100}$ is divided by $7$.

Please brief about the concept behind this to solve such problems. Thanks.

Using Euler-Fermat's theorem.
$\phi(7)=6$
$2^{6} \equiv 1 (\mod 7) \implies2^4.2^{96} \equiv 2(\mod7)$
$3^{6} \equiv 1 (\mod 7) \implies3^4.3^{96} \equiv 4(\mod7)$
$4^{6} \equiv 1 (\mod 7) \implies4^4.4^{96} \equiv 4(\mod7)$
$5^{6} \equiv 1 (\mod 7) \implies5^4.5^{96} \equiv 2(\mod7)$
$2^{100}+3^{100}+4^{100}+5^{100} \equiv 5(\mod 7)$
