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.)


We have

(\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}),

since integer multiplication is associative.

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