Find the remainder which is obtained when the number $\displaystyle 10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ is divided by 7.
By fermat's little theorem we know that
$\displaystyle 10^6 \equiv 1 \mod (7)$
so $\displaystyle 10^{10} = 10^6\cdot 10^4=10^4=1000$
By long divison this is $\displaystyle 4 \mod 7$
so $\displaystyle 10^{10}=4\mod(7)$
and note that
$\displaystyle 10 = 4 \mod (6)$
$\displaystyle 100 = 4 \mod (6)$
$\displaystyle 1000 = 4 \mod (6)$
ect ...
so all of the exponents are 4 mod 6
so with these two facts we get
$\displaystyle 10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4= $
but
$\displaystyle 10^4=4 \mod (7)$
$\displaystyle 10(4)=40$
now mod 7 we get 5