Elementary Methods in Number Theory Exercise 1.4.2

Factor $10^k+1$ into a product of primes for $k=1,2,3,4,5$.

Solve:

When $k=1$,
\begin{equation}
11=11
\end{equation}.
When $k=2$,
\begin{equation}
101=101
\end{equation}
When $k=3$,
\begin{equation}
1001=13\times 7\times 11
\end{equation}
When $k=4$,
\begin{equation}
10001=73\times 137
\end{equation}
When $k=5$,
\begin{equation}
100001=11\times 9091
\end{equation}