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## Multiplication of residue classes of integers is associative

Prove that multiplication of residue classes in $\mathbb{Z}/(n)$ is associative. (You may assume that it is well defined.)

Solution: We have \begin{align*} (\overline{a} \cdot \overline{b}) \cdot \overline{c} = &\ \overline{a \cdot b} \cdot \overline{c}\\ = &\ \overline{(a \cdot b) \cdot c}\\ = &\ \overline{a \cdot (b \cdot c)}\\ = &\ \overline{a} \cdot \overline{b \cdot c}\\ = &\ \overline{a} \cdot (\overline{b} \cdot \overline{c}), \end{align*} since integer multiplication is associative.