Question 42
Let $n \in \mathbb{N}$, $n \ge 3$. For any $x \in U(\mathbb{Z}_n)$ we denote $o_1(x)$ and $o_2(x)$ the orders of the element $x$ in the groups $(\mathbb{Z}_n, +)$, respectively $(U(\mathbb{Z}_n), \cdot)$, and $a(x) = o_1(x) + o_2(x)$. We consider the set $T_n = \{a(x) \mid x \in U(\mathbb{Z}_n)\}$.
a) Determine $T_8$.
b) Prove that there are at most 19 numbers $n$, for which $T_n$ has precisely two elements.
a) Determine $T_8$.
b) Prove that there are at most 19 numbers $n$, for which $T_n$ has precisely two elements.