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Addition of residue classes of integers is associative

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

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


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