15.1. $2{ }^{\star}$ Find the remainder when $10^{10}+10^{10^{2}}+10^{10^{3}}+\cdots+10^{10^{10}}$ is divided by 7.
It is known that $10^{6} \equiv 1(\bmod 7)$, so $10^{10} \equiv 10^{4}, 10^{10^{2}} \equiv 10^{4 \times 10} \equiv 10^{4}, 10^{10^{3}} \equiv$
$$
\begin{array}{l}
10^{4}, \cdots, 10^{10^{10}} \equiv 10^{4}(\bmod 7) \\
10^{10}+10^{10^{2}}+10^{10^{3}}+\cdots+10^{10^{10}} \equiv 10 \times 10^{4} \equiv 5(\bmod 7) .
\end{array}
$$