Algebra Solutions: The units in ZZ/(n) are closed under multiplication

Prove that if $\overline{a}, \overline{b} \in (\mathbb{Z}/(n))^\times$ , then $\overline{a} \cdot \overline{b} \in (\mathbb{Z}/(n))^\times$ .

Bars denote arithmetic mod n. Let $\overline{a}, \overline{b} \in (\mathbb{Z}/(n))^\times$ ; by the definition of $(\mathbb{Z}/(n))^\times$ , there exist $\overline{c}$ and $\overline{d}$ such that $\overline{a} \cdot \overline{c} = \overline{1}$ and $\overline{b} \cdot \overline{d} = \overline{1}$ . Thus we have $(\overline{a} \cdot \overline{b}) \cdot (\overline{d} \cdot \overline{c}) = \overline{a} \cdot \overline{1} \cdot \overline{c} = \overline{1}$ . Hence $\overline{a} \cdot \overline{b} \in (\mathbb{Z}/(n))^\times$ .

Algebra Solutions

The units in ZZ/(n) are closed under multiplication

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