4.48. Find the remainder when the number

$$
10^{10}+10^{\left(10^{2}\right)}+10^{\left(10^{3}\right)}+\ldots+10^{\left(10^{10}\right)}
$$

is divided by 7.
4.48. Answer: 5. Note that $10^{6} \equiv 1(\bmod 7)$, since $10^{3}+1$ is divisible by 7, and $10^{k} \equiv 4(\bmod 6)$ for $k \geqslant 1$, since the number $99 \ldots 96$ is even and divisible by 3. Therefore, $10^{10^{k}} \equiv 10^{4}(\bmod 7)$ for $k \geqslant 1$. Hence, the required remainder is the remainder of the division of the number $10 \cdot 10^{4}=10^{5}$ by 7. This remainder is 5.