Definition:Modulo Addition

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Definition

Definition 1

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \left\{{\left[\!\left[{0}\right]\!\right]_m, \left[\!\left[{1}\right]\!\right]_m, \ldots, \left[\!\left[{m-1}\right]\!\right]_m}\right\}$

where $\left[\!\left[{x}\right]\!\right]_m$ is the residue class of $x$ modulo $m$.


The operation of addition modulo $m$ is defined on $\Z_m$ as:

$\left[\!\left[{a}\right]\!\right]_m +_m \left[\!\left[{b}\right]\!\right]_m = \left[\!\left[{a + b}\right]\!\right]_m$


Definition 2

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$


The operation of addition modulo $m$ is defined on $\Z_m$ as:

$x +_m y$ equals the remainder after $x + y$ has been divided by $m$.


Definition 3

Let $m \in \Z$ be an integer.

Let $\Z_m$ be the set of integers modulo $m$:

$\Z_m = \left\{{0, 1, \ldots, m-1}\right\}$


The operation of addition modulo $m$ is defined on $\Z_m$ as:

$x +_m y := x + y - j m$

where $j$ is the largest integer such that $j m \le x + y$.


Also denoted as

Although the operation of addition modulo $z$ is denoted by the symbol $+_z$, if there is no danger of confusion, the symbol $+$ is often used instead.


The notation for addition of two residue classes modulo $z$ is not usually $\left[\!\left[{a}\right]\!\right]_z +_z \left[\!\left[{b}\right]\!\right]_z$.

What is more normally seen is $a + b \pmod z$.


Cayley Table

The additive group of integers modulo $m$ can be described by showing its Cayley table.


This one is for modulo $6$:

$\begin{array}{r|rrrrrr} \left({\Z_6, +_6}\right) & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 \\ \hline \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 \\ \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 \\ \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 \\ \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 \\ \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 \\ \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{5}\right]\!\right]_6 & \left[\!\left[{0}\right]\!\right]_6 & \left[\!\left[{1}\right]\!\right]_6 & \left[\!\left[{2}\right]\!\right]_6 & \left[\!\left[{3}\right]\!\right]_6 & \left[\!\left[{4}\right]\!\right]_6 \\ \end{array}$

which can also be presented:

$\begin{array}{r|rrrrrr} +_m & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 1 & 2 & 3 & 4 & 5 \\ 1 & 1 & 2 & 3 & 4 & 5 & 0 \\ 2 & 2 & 3 & 4 & 5 & 0 & 1 \\ 3 & 3 & 4 & 5 & 0 & 1 & 2 \\ 4 & 4 & 5 & 0 & 1 & 2 & 3 \\ 5 & 5 & 0 & 1 & 2 & 3 & 4 \\ \end{array}$


Also see