Let \(p\) be an odd prime and \(\alpha \in \mathbb{Z_p^*}\). Let \(\mathcal{P} := \{ S \subset \mathbb{Z_p^*}: \lvert S \rvert = 2\}\), and define \(\mathcal{C} := \{ \{\kappa, \lambda\} \in \mathcal{P} : \kappa\lambda = \alpha\} \). Note that for every \( \kappa \in \mathbb{Z_p^*}\), there is a unique \(\lambda \in \mathbb{Z_p^*}\) such that \( \kappa \lambda = \alpha \), namely \(\displaystyle \lambda := \frac{\alpha}{\kappa} \); moreover, \(\kappa \ne \lambda\). Define \(\mathcal{D} := \{ \{\kappa, \lambda\} \in \mathcal{P} : \kappa\lambda = 1\} \). Calculate the sets \(\mathcal{C}\) and \(\mathcal{D}\) in the case \(p = 11\) and \(\alpha = -1\).

\(\mathcal{C}: \{\{1,10\}, \{2,5\}, \{3,7\}, \{4,8\}, \{6, 9\}\}\)
\(\mathcal{D}: \{\{2,6\}, \{3,4\}, \{5,9\}, \{7,8\}\}\)

# posted by rot13(Unafba Pune) @ 10:13 PM