\[\begin{array}{l} N = \overline {abc} = 7\overline {ac} \Leftrightarrow \hfill \\ 100a + 10b + c = 7 \cdot \left( {10a + c} \right) \Leftrightarrow 15a + 5b = 3c \Rightarrow \left\{ \begin{array}{l} c = 5 \hfill \\ a = 1 \hfill \\ b = 0 \hfill \\ \end{array} \right. \hfill \\ N = 105 \hfill \\ \end{array}\]

105