1. ## Find the remainder?

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.

2. Originally Posted by fardeen_gen
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

3. Hello TheEmptySet,
Thank you! Is it possible to do this by binomial theorem too?

4. Originally Posted by fardeen_gen
Hello TheEmptySet,
Thank you! Is it possible to do this by binomial theorem too?
I don't see a way, but you never know.