\[Пусть\ \text{x\ }и\ y - два\ натуральных\ числа.\]
\[x \cdot y = 154;\]
\[x^{2} + y^{2} = 317.\]
\[Составим\ систему\ уравнений:\]
\[\left\{ \begin{matrix} xy = 154\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ x^{2} + y^{2} = 317\ \ \ \ \ \ \\ \end{matrix} \right.\ \ \left\{ \begin{matrix} x = \frac{154}{y}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \left( \frac{154}{y} \right)^{2} + y^{2} = 317 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ }\]
\[y^{4} - 317y^{2} + 154^{2} = 0\]
\[D = 317^{2} - 4 \cdot 154^{2} =\]
\[= 100\ 489 - 94\ 864 = 5\ 625\]
\[y^{2} = \frac{317 + \sqrt{5625}}{2} = \frac{317 + 75}{2} = \frac{392}{2} = 196\ \]
\[y = \pm \sqrt{196} = \pm 16;\]
\[y^{2} = \frac{317 - \sqrt{5625}}{2} = \frac{317 - 95}{2} = \frac{242}{2} = 121\]
\[y = \pm \sqrt{121} = \pm 11\]
\[\left\{ \begin{matrix} y = 14 \\ x = 11 \\ \end{matrix} \right.\ \text{\ \ \ \ \ \ \ \ }\left\{ \begin{matrix} y = 11 \\ x = 14 \\ \end{matrix} \right.\ \]
\[Ответ:два\ числа\ 11\ и\ 14.\]