Find the remainder?

**fardeen_gen** · May 2nd 2009, 09:15 PM

Find the remainder which is obtained when the number $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ is divided by 7.

**TheEmptySet** · May 2nd 2009, 10:11 PM

Originally Posted by fardeen_gen

Find the remainder which is obtained when the number $10^{10} + 10^{10^2} + 10^{10^3} + ... + 10^{10^{10}}$ is divided by 7.

By fermat's little theorem we know that

$10^6 \equiv 1 \mod (7)$

so $10^{10} = 10^6\cdot 10^4=10^4=1000$

By long divison this is $4 \mod 7$

so $10^{10}=4\mod(7)$

and note that

$10 = 4 \mod (6)$
$100 = 4 \mod (6)$
$1000 = 4 \mod (6)$

ect ...

so all of the exponents are 4 mod 6

so with these two facts we get

$10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4+10^4=$
but
$10^4=4 \mod (7)$
$10(4)=40$

now mod 7 we get 5

**fardeen_gen** · May 2nd 2009, 10:16 PM

Hello TheEmptySet,
Thank you!

Is it possible to do this by binomial theorem too?

**TheEmptySet** · May 2nd 2009, 10:19 PM

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.

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