Find $\overline{0},\overline{1},\overline{10}$ and $\overline{16}$ in $\mathbb{Z}_5$

Question

Find $\overline{0},\overline1,\overline{10}$ and $\overline{16}$ in $\mathbb{Z}_5$

I know that the bar above the number means the congruence modulo.

$\overline{a}:=\{x\in \mathbb{Z}:x\equiv a \pmod n\}$. And I know that $x\equiv a \pmod n$ if $n \mid (x-a)$.

From what I can understand, $\mathbb{Z}_5 = \{0,1,2,3,4\}$.

But I'm not sure how to connect these concepts.

Answer 1

Saying $\mathbb{Z}_5=\{0,1,2,3,4\}$ is just a bit of abuse of lenguaje for $\mathbb{Z}_5=\{\bar0,\bar1,\bar2,\bar3,\bar4\}$. Once you explicitly talk about cosets its obligated to denote $\mathbb{Z}_5=\{\bar0,\bar1,\bar2,\bar3,\bar4\}$.

Now to find that $\overline{16}=\bar1$ just need to notice that $16=1\mod 5$. In the same way you have that $\overline{10}=\bar0$ and, of course that $\bar0=\bar0$ and $\bar1=\bar1$.

Answer 2

Note that $\mathbb{Z}_5$ is actually the set of congruence classes modulo $5$, so in your notation: $$\mathbb{Z}_5=\{\overline{0},\overline{1}, \overline{2}, \overline{3},\overline{4}\} $$ Thus we have $\overline{10}=\overline{0} \in \mathbb{Z}_5$ and $\overline{16}=\overline{1}\in \mathbb{Z}_5$.

Answer 3

the ${\bar 0}$ in $\Bbb{Z}_5$ is the class of $0$ modulo 5 in $\Bbb{Z}$, that is the sub-set of $\Bbb{Z}$ defined as ${\bar 0}=\{5n, \; n\in \Bbb{Z}\}$, also ${\bar 1}=\{5n+1, \; n\in \Bbb{Z}\}$, so ${\bar 0}={\bar {10}}$ and ${\bar 1}={\bar {16}}$