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

#
$10^{6} \equiv 1(\bmod 7)$, since $10^{3}+1$ is divisible by 7, and $10^{k} \equiv 4(\bmod 6)$ for $k \geq 1$, since the number $9 \ldots . .96$ is even and divisible by 3. Therefore,

## Methods for solving problems with parameters

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$10^{10^{k}} \equiv 10^{4}(\bmod 7)$ for $k \geq 1$. Therefore, the required remainder is the remainder of the division of the number $10 \cdot 10^{4} \equiv 3^{5} \equiv 2$

$(\bmod 7)$.

## Answer

5.