Exercise 1.1.3 (Associativity of addition in $\mathbb{Z}/n\mathbb{Z}$)

Question

Prove that addition of residues classes in $\mathbb{Z}/n\mathbb{Z}$ is associative.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} 
Let $A, B, C$ be elements of $\Z/n\Z$. There are integers $a, b ,c$ such that
$$A = \overline{a},\qquad B = \overline{b},\qquad C = \overline{c}.$$
By definition of addition in $\mathbb{Z}/n\mathbb{Z}$,
\begin{align*}
(A + B) + C &= (\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})\
&= A + (B+C).
\end{align*}
The addition of residues classes in $\mathbb{Z}/n\mathbb{Z}$ is associative.
\end{proof}

Exercise 1.1.3 (Associativity of addition in $\mathbb{Z}/n\mathbb{Z}$)

Answers

Comments

Exercise 1.1.3 (Associativity of addition in $\mathbb{Z}/n\mathbb{Z}$)

Answers

Comments

Add answer