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\noindent{\Large{\textsc{Number Theory \hfill Assignment 5.3  \\ Fall 2023 \hfill Due September 27}}}

\large

\vspace{1cm}


Recall that for $n \in \N$ we defined
\[
\Z /n \Z = \{ a + n \Z \, | \, a \in \Z\} = \{n\Z , 1 + n \Z , 2+ n\Z , \ldots, (n-1) + n\Z \},
\]
which is the set of congruence classes modulo $n$.

\vspace{0.5cm}

\begin{exercise}
Fix $n \in \N$ and let $a_1, a_2 , \ldots, a_n \in \Z$.  Prove that the following are equivalent.
\begin{itemize}

\item[\bf (i)]  The integers $a_1, a_2 , \ldots, a_n$ form a complete residue system modulo $n$.

\item[\bf (ii)]  For any $a \in \Z$ there exists an $i$ so that $a_i \equiv a \pmod{n}$.  [{\em Question.}  How does this differ from the actual definition of ``complete residue system modulo $n$?'']

\item[\bf (iii)]  For all $1 \le i,j \le n$:
\[
i \ne j \,\,\, \Rightarrow \,\,\, a_i \not \equiv a_j \pmod{n}. 
\]

\item[\bf (iv)]  $\Z /n\Z = \{a_1 + n\Z, a_2 + n\Z , \ldots, a_n + n\Z \}$ (in this case one says that $a_1, a_2 , \ldots, a_n$ {\em represent} $\Z/n\Z$).

\end{itemize}

\noindent 

\end{exercise}

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\begin{exercise}
Textbook exercise 4.2.12.
\end{exercise}

\begin{exercise}
Textbook exercise 4.2.15.
\end{exercise}

\begin{exercise}
Textbook exercise 4.2.16.
\end{exercise}

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