Exercise 1.1.4 (Associativity of multiplication in $\mathbb{Z}/n\mathbb{Z}$)

Question

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

Richard Ganaye · Accepted Answer

\begin{proof} Similarly, for
$$A = \overline{a},\qquad B = \overline{b},\qquad C = \overline{c}.$$
By definition of multiplication in $\mathbb{Z}/n\mathbb{Z}$,
\begin{align*}
(A \cdot B) \cdot C &= (\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})\
&= A \cdot (B\cdot C).
\end{align*}
The addition of residues classes in $\mathbb{Z}/n\mathbb{Z}$ is associative.
\end{proof}

Exercise 1.1.4 (Associativity of multiplication in $\mathbb{Z}/n\mathbb{Z}$)

Answers

Comments

Exercise 1.1.4 (Associativity of multiplication in $\mathbb{Z}/n\mathbb{Z}$)

Answers

Comments

Add answer