\[\begin{array}{l} {\text{1 способ}}\\ 1 + 13 + {13^2} + ... + {13^{2007}} = \\ \left( {1 + 13} \right) + \left( {{{13}^2} + {{13}^3}} \right) + ... + \left( {{{13}^{2006}} + {{13}^{2007}}} \right)\\ \end{array}\]

\[\begin{array}{l} 2{\text{ способ}}\\ {\text{Заметим}}{\text{, что}}\\ 1 + 13 + {13^2} + ... + {13^{2007}} = \frac{{{{13}^{2008}} - 1}}{{13 - 1}} \end{array}\]

\[\begin{array}{l} 1 + 13 + {13^2} + ... + {13^{2007}} = \\ \left( {1 + 13} \right) + \left( {{{13}^2} + {{13}^3}} \right) + ... + \left( {{{13}^{2006}} + {{13}^{2007}}} \right) = \\ 14 + 14 \cdot {13^2} + ... + 14 \cdot {13^{2006}} = \\ 14 \cdot \left( {1 + {{13}^2} + ... + {{13}^{2006}}} \right) \Rightarrow \\ {\text{это число делится на 7}}{\text{.}} \end{array}\]