\[\boxed{\mathbf{1043}\mathbf{.}\mathbf{\ }}\]
\[1)\ a + \frac{1}{a} = 7\frac{1}{7}\]
\[7 + \frac{1}{7} = 7\frac{1}{7}\]
\[\Longrightarrow a = 7.\]
\[2)\ b - \frac{1}{b} = 14\frac{14}{15}\]
\[15 - \frac{1}{15} = 14\frac{14}{15}\]
\[\Longrightarrow b = 15.\]