Section 2.2: Modular Arithmetic

Given any set $\mathbb{Z}_n$ , where $n\in\mathbb{Z}$ we define two operations: $\oplus$ and $\odot$ .

$\begin{equation}\label{ma_plus} [a] \oplus [b] = [ a+b ] \end{equation}$

and

$\begin{equation}\label{ma_dot}[a] \odot [b] = [ab]\end{equation}$

Modular Arithmetic is a binary operation. That is, a binary operation is where there are two inputs and one output.

Example in $\mathbb{Z}_7$ :
$\begin{align*}[3]\oplus [5] &=[8]=[1] \\ [10]\oplus [12] &=[22]=[1] \\ [3] \odot [5] &= [15]=[1] \\ [10] \odot [12] &= [120]=[1]\end{align*}$

Theorem 2.6

If $[a]=[b]$ and $[c]=[d]$ in $\mathbb{Z}_n$ , then $[a+c]=[bdd]$ and $[ac]=[bd]$ .

Example Modular Arithmetic table in $\mathbb{Z}_3$ :

Modular Arithmetic has several different types of properties.

Theorem 2.7

$[a]\oplus [b] \in \mathbb{Z} \mbox{ (closure under addition) }$

$[a]\oplus ([b]\oplus [c] = ([a]\oplus [b]) \oplus [c] \mbox{ (associative) }$

$[a]\oplus [b] = [b] \oplus [a] \mbox{ (commutative) }$

$[a]\oplus [0] = [a] = [0]\oplus [a] \mbox{ (additive identity) }$

$\mbox{ For each } [a]\in \mathbb{Z}\mbox{, the equation }[a]\oplus x = [0] \mbox{ has a solution.}$

$[a]\oplus [b] \in \mathbb{Z}_n \mbox{ (closure under multiplication)}$

$[a]\odot ([b]\odot [c])=([a]\odot [b])\odot [c] \mbox{ (associative identity)}$

$[a]\odot ([b]\oplus [c])=[a]\odot [b] \oplus [a]\odot [c] \mbox{ which is equivalent to the expression } ([a]\oplus [b])\odot [c] = [a]\odot [c] \oplus [b]\odot [c]$

$[a]\odot [b]=[b]\odot [a]$

$[a] \odot [1]=[a]=[1]\odot [a]$

Powers

We can further this by putting a congruence set $[a]$ to a power. Say $[a]^n=[a]\odot [a]\odot \ldots \odot [a]$ . Jut as with exponents with integers, there are n number of $[a]$ being multiplied.

A Power Example

For $\mathbb{Z}_10$ we have $[3]^4=[81]=1$ .

General Problems

$\begin{align*}\mbox{For } \mathbb{Z}_5 & & x^2\oplus [1]=[0] \\ x=[0] & & x^2\oplus [1]=[1]\ne[0] \\ x=[1]& & x^2\oplus [1]=[2]\ne [0]\\ x=[2]& & x^2\oplus [1]=[5]\\ x=[3]& & x^2\oplus [1]=[10]\\ x=[4]& & x^2\oplus [1]=[17]=[2]\ne [0] \end{align*}$