nyuuzyou/wikis huggingface dataset:
Modulo Addition has Inverses

== Theorem ==
Let $m \in \Z$ be an integer.
Then addition modulo $m$ has inverses:
For each element $\eqclass x m \in \Z_m$, there exists the element $\eqclass {-x} m \in \Z_m$ with the property:
:$\eqclass x m +_m \eqclass {-x} m = \eqclass 0 m = \eqclass {-x} m +_m \eqclass x m$
where $\Z_m$ is the set of integers modulo $m$.
That is:
:$\forall a \in \Z: a + \paren {-a} \equiv 0 \equiv \paren {-a} + a \pmod m$
== Proof ==
Let $x \in \Z$ be arbitrary.
As $x \in \Z$ implies $-x \in \Z$, we have:
:$\eqclass {-x} m \in \Z_m$
Therefore:
== Sources ==
* : Chapter $\text{II}$: Groups: A Little Number Theory
* : $\S 1$: Some examples of groups: Example $1.10$
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Addition
Modulo Addition is Associative

== Theorem ==
Addition modulo $m$ is associative:
:$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m +_m \eqclass y m} +_m \eqclass z m = \eqclass x m +_m \paren {\eqclass y m +_m \eqclass z m}$
where $\Z_m$ is the set of integers modulo $m$.
That is:
:$\forall x, y, z \in \Z: \paren {x + y} + z \equiv x + \paren {y + z} \pmod m$
== Proof ==
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.6$
* : $\S 1$: Some examples of groups: Example $1.10$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Addition
Modulo Addition is Closed

== Theorem ==
=== Integers ===
=== Real Numbers ===
== Note ==
On the face of it, there seems no need to prove this for both integers and reals.
From the above we see that the sum of two integers congruent to a given integer modulus is congruent to an integer to the same modulus.
However, it is important to note that the sum of two integers congruent to a given real modulus may not be congruent to an integer.
Take, as an example, $a = 2, b = 3, m = \pi$.
Then $\left({a + b}\right) \pmod \pi = 5 - \pi \notin \Z$.
Category:Modulo Addition
Modulo Addition is Closed/Integers

== Theorem ==
Let $m \in \Z$ be an integer.
Then addition modulo $m$ on the set of integers modulo $m$ is closed:
:$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m \in \Z_m$.
== Proof ==
From the definition of addition modulo $m$, we have:
:$\eqclass x m +_m \eqclass y m = \eqclass {x + y} m$
By the Division Theorem:
:$x + y = q m + r$ where $0 \le r < m$
Therefore for all $0 \le r < m$:
:$\eqclass {x + y} m = \eqclass r m$.
Therefore from the definition of integers modulo $m$:
:$\eqclass {x + y} m \in \Z_m$
== Also see ==
* Modulo Addition is Closed/Real Numbers
== Sources ==
* : $\S 1$: Some examples of groups: Example $1.10$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Addition
Modulo Addition is Closed/Real Numbers

== Theorem ==
Let $z \in \R$ be a real number.
Then addition modulo $z$ on the set of residue classes modulo $z$ is closed:
:$\forall \eqclass x z, \eqclass y z \in \R_z: \eqclass x z +_z \eqclass y z \in \R_z$.
== Proof ==
From the definition of addition modulo $z$, we have:
:$\eqclass x z +_z \eqclass y z = \eqclass {x + y} z$
As $x, y \in R$, we have that $x + y \in \R$ as Real Addition is Closed.
Hence by definition of congruence, $\eqclass {x + y} z \in \R_z$.
== Also see ==
* Modulo Addition is Closed/Integers
Category:Modulo Addition
Modulo Addition is Commutative

== Theorem ==
Modulo addition is commutative:
:$\forall x, y, z \in \Z: x + y \pmod m = y + x \pmod m$
== Proof ==
From the definition of modulo addition, this is also written:
:$\forall m \in \Z: \forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m +_m \eqclass y m = \eqclass y m +_m \eqclass x m$
By Modulo Addition is Well-Defined, $+_m$ is well-defined.
Hence:
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.7$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Addition
Modulo Addition is Linear

== Theorem ==
Let $m \in \Z_{> 0}$.
Let $x_1, x_2, y_1, y_2, c_1, c_2 \in \Z$.
Let:
:$x_1 \equiv y_1 \pmod m$
:$x_2 \equiv y_2 \pmod m$
Then:
:$c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$
== Proof ==
By Scaling preserves Modulo Addition:
:$c_1 x_1 \equiv c_1 y_1 \pmod m$
:$c_2 x_2 \equiv c_2 y_2 \pmod m$
and so by Modulo Addition is Well-Defined:
:$c_1 x_1 + c_2 x_2 \equiv c_1 y_1 + c_2 y_2 \pmod m$
== Sources ==
* : Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)} \ \text{(A)}$
Category:Modulo Addition
Modulo Addition is Well-Defined

== Theorem ==
Let $m \in \Z$ be an integer.
Let $\Z_m$ be the set of integers modulo $m$.
Let $\eqclass a m$ denote the equivalence class on $\Z_m$, for some $a \in \Z$.
The modulo addition operation on $\Z_m$, defined by the rule:
:$\eqclass a m +_m \eqclass b m = \eqclass {a + b} m$
is a well-defined operation.
That is:
:If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a + x \equiv b + y \pmod m$.
=== Corollary ===
It follows that:
=== Real Modulus ===
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Sources ==
* : Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
* : Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $2$
* : Chapter $1$: Equivalence Relations: $\S 19 \beta$
* : $\S 14.3 \ \text {(i)}$: Congruence modulo $m$ ($m \in \N$)
* : $\S 2.3$: Congruences: Theorem $1 \ \text{(i)}$
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{A}$
* 
Category:Modulo Addition is Well-Defined
Category:Modulo Addition
Modulo Addition is Well-Defined/Examples

== Examples of Use of Modulo Addition is Well-Defined ==
=== Modulo Addition: $2 + 3 \equiv -6 + 15 \pmod 4$ ===
=== Modulo Addition: $19 + 6 \equiv 11 + 2 \pmod 4$ ===
Category:Examples of Modulo Arithmetic
Modulo Addition is Well-Defined/Examples/2+3 equiv -6+15 mod 4

== Example of Use of Modulo Addition is Well-Defined ==
We have:
To confirm:
That is:
:$\eqclass 2 4 + \eqclass 3 4 = \eqclass 1 4$
== Sources ==
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Example $8$
Category:Examples of Modulo Arithmetic
Modulo Addition is Well-Defined/Proof 1

== Theorem ==
== Proof ==
We need to show that if:
:$\eqclass {x'} m = \eqclass x m$
:$\eqclass {y'} m = \eqclass y m$
then:
:$\eqclass {x' + y'} m = \eqclass {x + y} m$
Since:
:$\eqclass {x'} m = \eqclass x m$
and:
:$\eqclass {y'} m = \eqclass y m$
it follows from the definition of set of integers modulo $m$ that:
:$x \equiv x' \pmod m$
and:
:$y \equiv y' \pmod m$
By definition, we have:
:$x \equiv x' \pmod m \implies \exists k_1 \in \Z: x = x' + k_1 m$
:$y \equiv y' \pmod m \implies \exists k_2 \in \Z: y = y' + k_2 m$
which gives us:
:$x + y = x' + k_1 m + y' + k_2 m = x' + y' + \paren {k_1 + k_2} m$
As $k_1 + k_2$ is an integer, it follows that, by definition:
:$x + y \equiv \paren {x' + y'} \pmod m$
Therefore, by the definition of integers modulo $m$:
:$\eqclass {x' + y'} m = \eqclass {x + y} m$
== Sources ==
* : Chapter $\text{II}$: Groups: A Little Number Theory
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $4$
* : $\text {4-1}$ Basic Properties of Congruences: Theorem $\text {4-2}$
* : $\S 2.3$: Congruences
* : Chapter $2$: Maps and relations on sets: Proposition $2.31$
Category:Modulo Addition is Well-Defined
Modulo Addition is Well-Defined/Proof 2

== Theorem ==
== Proof ==
The equivalence class $\eqclass a m$ is defined as:
:$\eqclass a m = \set {x \in \Z: x = a + k m: k \in \Z}$
That is, the set of all integers which differ from $a$ by an integer multiple of $m$.
Thus the notation for addition of two set of integers modulo $m$ is not usually:
:$\eqclass a m +_m \eqclass b m$
What is more normally seen is:
:$a + b \pmod m$
Using this notation:
== Sources ==
* : $\S 2.6$. Algebra of congruences: $\text{(i)}$
* : Chapter $\text {II}$: New Structures from Old: $\S 11$: Quotient Structures: Example $11.2$
* : $\S 1$: Some examples of groups: Example $1.10$
* : $\S 18.4$: Congruence classes
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $17$
Category:Modulo Addition is Well-Defined
Modulo Addition is Well-Defined/Real Modulus

== Theorem ==
Let $z \in \R$ be a real number.
Let:
:$a \equiv b \pmod z$
and:
:$x \equiv y \pmod z$
where $a, b, x, y \in \R$.
Then:
: $a + x \equiv b + y \pmod z$
== Proof ==
== Sources ==
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $24$
Category:Modulo Addition
Category:Modulo Addition is Well-Defined
Modulo Arithmetic/Examples

== Examples of Modulo Arithmetic ==
=== $11$ divides $3^{3 n + 1} + 2^{2 n + 3}$ ===
=== Solutions to $x^2 = x \pmod 6$ ===
=== Multiplicative Inverse of $41 \pmod {97}$ ===
=== Residue of $2^{512} \pmod 5$ ===
=== $n \paren {n^2 - 1} \paren {3 n - 2}$ Modulo $24$ ===
=== $a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1$ Modulo $12$ ===
Category:Examples of Modulo Arithmetic
Modulo Arithmetic/Examples/11 Divides 3^3n+1 + 2^2n+3

== Example of Modulo Arithmetic ==
:$11$ is a divisor of $3^{3 n + 1} + 2^{2 n + 3}$.
== Proof ==
We have:
Now:
Then we have:
and:
So:
Hence:
:$\forall n \in \N: 11 \divides \paren {3^{3 n + 1} + 2^{4 n + 3} }$
== Sources ==
* : Chapter $2$: Some Properties of $\Z$: Exercise $2.10$
Category:Examples of Modulo Arithmetic
Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97

== Example of Modulo Arithmetic ==
The inverse of $41$ under multiplication modulo $97$ is given by:
:${\eqclass {41} {97} }^{-1} = 71$
=== Solution to $41 x \equiv 2 \pmod {97}$ ===
== Proof ==
From Ring of Integers Modulo Prime is Field, multiplication modulo $97$ has an inverse for all $x \in \Z_{97}$ where $x \ne 0$.
Using Euclid's Algorithm:
Then:
So:
: $\paren {-26} \times 41 \equiv 1 \pmod {97}$
Hence the result.
== Sources ==
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $8$
Category:Examples of Modulo Arithmetic
Category:Multiplicative Inverse of 41 Modulo 97
Modulo Arithmetic/Examples/Multiplicative Inverse of 41 Modulo 97/41x = 2 Modulo 97

== Example of Modulo Arithmetic ==
The solution of the integer congruence:
:$41 x \equiv 2 \pmod {97}$
is:
:$x = 45$
== Proof ==
From Multiplicative Inverse of $41 \pmod {97}$ we have:
:$41 \times 71 = 1 \pmod {97}$
Thus:
Hence the result.
== Sources ==
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $8$
Category:Multiplicative Inverse of 41 Modulo 97
Modulo Arithmetic/Examples/Solutions to x^2 = x Modulo 6

== Example of Modulo Arithmetic ==
The equation:
:$x^2 = x \pmod 6$
has solutions:
== Proof ==
The Cayley table for multiplication modulo $6$ can be presented as:
:$\begin{array} {r|rrrrrr}
\times_6 & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
0 & 0 & 0 & 0 & 0 & 0 & 0 
\\
1 & 0 & 1 & 2 & 3 & 4 & 5
\\
2 & 0 & 2 & 4 & 0 & 2 & 4
\\
3 & 0 & 3 & 0 & 3 & 0 & 3
\\
4 & 0 & 4 & 2 & 0 & 4 & 2
\\
5 & 0 & 5 & 4 & 3 & 2 & 1
\\
\end{array}$
The squares $x^2$ of each $x$ can be found on the main diagonal, where each element of $\Z_6$ can be inspected.
Checking each of these, we have:
== Sources ==
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $7$
Category:Examples of Modulo Arithmetic
Modulo Arithmetic/Examples/a^2 + (a+2)^2 + (a+4)^2 + 1 Modulo 12

== Example of Modulo Arithmetic ==
Let $a$ be an odd integer.
Then:
:$a^2 + \paren {a + 2}^2 + \paren {a + 4}^2 + 1 \equiv 0 \pmod {12}$
== Proof ==
Let $a = 2 k + 1$ where $k \in \Z$.
Then:
Hence the result.
== Sources ==
* : $\text {2-1}$ Euclid's Division Lemma: Exercise $7$
Category:Examples of Modulo Arithmetic
Modulo Arithmetic/Examples/n(n^2-1)(3n-2) Modulo 24

== Example of Modulo Arithmetic ==
:$n \paren {n^2 - 1} \paren {3 n + 2} \equiv 0 \pmod {24}$
== Proof ==
The proof proceeds by induction.
For all $n \in \Z_{> 0}$, let $\map P n$ be the proposition:
:$n \paren {n^2 - 1} \paren {3 n + 2} \equiv 0 \pmod {24}$
=== Basis for the Induction ===
$\map P 1$ is the case:
Thus $\map P 1$ is seen to hold.
This is the basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
:$k \paren {k^2 - 1} \paren {3 k + 2} \equiv 0 \pmod {24}$
from which it is to be shown that:
:$\paren {k + 1} \paren {\paren {k + 1}^2 - 1} \paren {3 \paren {k + 1} + 2} \equiv 0 \pmod {24}$
=== Induction Step ===
This is the induction step:
By the induction hypothesis:
:$k \paren {k^2 - 1} \paren {3 k + 2} = 24 r$
for some $r \in \Z$.
Take $12 k \paren {k + 1}^2$.
If $k$ is even, then $12 k$ and so $12 k \paren {k + 1}^2$ is divisible by $24$.
If $k$ is odd, then $k + 1$ is even and so $12 k \paren {k + 1}^2$ is again divisible by $24$.
Thus:
:$12 k \paren {k + 1}^2 = 24 s$
for some $s \in \Z$.
Thus:
:$k \paren {k^2 - 1} \paren {3 k + 2} + 12 k \paren {k + 1}^2 = 24 \paren {r + s}$
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
:$\forall n \in \Z_{>0}: n \paren {n^2 - 1} \paren {3 n + 2} \equiv 0 \pmod {24}$
== Sources ==
* : $\text {1-1}$ Principle of Mathematical Induction: Exercise $18$
Category:Examples of Modulo Arithmetic
Modulo Multiplication/Cayley Table

== Cayley Table for Modulo Multiplication ==
The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.
=== Modulo 3 ===
=== Modulo 4 ===
=== Modulo 5 ===
=== Modulo 6 ===
Category:Examples of Cayley Tables
Category:Modulo Multiplication
Modulo Multiplication/Cayley Table/Modulo 3

== Cayley Table for Modulo Multiplication ==
The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.
This one is for modulo $3$:
:$\begin {array} {r|rrr} \struct {\Z_3, \times_3} & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 & \\ \hline \eqclass 0 3 & \eqclass 0 3 & \eqclass 0 3 & \eqclass 0 3\\\eqclass 1 3 & \eqclass 0 3 & \eqclass 1 3 & \eqclass 2 3 \\ \eqclass 2 3 & \eqclass 0 3 & \eqclass 2 3 & \eqclass 1 3 \\ \end {array}$
which can also be presented:
:$\begin {array} {r|rrrrr} \times_3 & 0 & 1 & 2 \\ \hline 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 \\ 2 & 0 & 2 & 1 \end {array}$
Category:Examples of Cayley Tables
Category:Modulo Multiplication
Modulo Multiplication/Cayley Table/Modulo 4

== Cayley Table for Modulo Multiplication ==
The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.
This one is for modulo $4$:
:$\begin {array} {r|rrrrr} \struct {\Z_4, \times_4} & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \hline \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 & \eqclass 0 4 \\ \eqclass 1 4 & \eqclass 0 4 & \eqclass 1 4 & \eqclass 2 4 & \eqclass 3 4 \\ \eqclass 2 4 & \eqclass 0 4 & \eqclass 2 4 & \eqclass 0 4 & \eqclass 2 4 \\ \eqclass 3 4 & \eqclass 0 4 & \eqclass 3 4 & \eqclass 2 4 & \eqclass 1 4 \\ \end {array}$
which can also be presented:
:$\begin {array} {r|rrrrr} \times_4 & 0 & 1 & 2 & 3 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 \\ 2 & 0 & 2 & 0 & 2 \\ 3 & 0 & 3 & 2 & 1 \\ \end {array}$
== Sources ==
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes
Category:Examples of Cayley Tables
Category:Modulo Multiplication
Modulo Multiplication/Cayley Table/Modulo 5

== Cayley Table for Modulo Multiplication ==
The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.
This one is for modulo $5$:
:$\begin {array} {r|rrrrr} \struct {\Z_5, \times_5} & \eqclass 0 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \hline \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 & \eqclass 0 5 \\ \eqclass 1 5 & \eqclass 0 5 & \eqclass 1 5 & \eqclass 2 5 & \eqclass 3 5 & \eqclass 4 5 \\ \eqclass 2 5 & \eqclass 0 5 & \eqclass 2 5 & \eqclass 4 5 & \eqclass 1 5 & \eqclass 3 5 \\ \eqclass 3 5 & \eqclass 0 5 & \eqclass 3 5 & \eqclass 1 5 & \eqclass 4 5 & \eqclass 2 5 \\ \eqclass 4 5 & \eqclass 0 5 & \eqclass 4 5 & \eqclass 3 5 & \eqclass 2 5 & \eqclass 1 5 \\ \end {array}$
which can also be presented:
:$\begin {array} {r|rrrrr} \times_5 & 0 & 1 & 2 & 3 & 4 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 \\ 2 & 0 & 2 & 4 & 1 & 3 \\ 3 & 0 & 3 & 1 & 4 & 2 \\ 4 & 0 & 4 & 3 & 2 & 1 \\ \end {array}$
== Sources ==
* : $\S 19$: Properties of $\Z_m$ as an algebraic system
* 
Category:Examples of Cayley Tables
Category:Modulo Multiplication
Modulo Multiplication/Cayley Table/Modulo 6

== Cayley Table for Modulo Multiplication ==
The multiplicative monoid of integers modulo $m$ can be described by showing its Cayley table.
This one is for modulo $6$:
$\quad \begin{array} {r|rrrrrr}
\struct {\Z_6, \times_6} & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6 \\
\hline
\eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 & \eqclass 0 6 
\\
\eqclass 1 6 & \eqclass 0 6 & \eqclass 1 6 & \eqclass 2 6 & \eqclass 3 6 & \eqclass 4 6 & \eqclass 5 6
\\
\eqclass 2 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6 & \eqclass 0 6 & \eqclass 2 6 & \eqclass 4 6
\\
\eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6 & \eqclass 0 6 & \eqclass 3 6
\\
\eqclass 4 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6 & \eqclass 0 6 & \eqclass 4 6 & \eqclass 2 6
\\
\eqclass 5 6 & \eqclass 0 6 & \eqclass 5 6 & \eqclass 4 6 & \eqclass 3 6 & \eqclass 2 6 & \eqclass 1 6
\end{array}$
which can also be presented:
$\quad \begin{array} {r|rrrrrr} \times_6 & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 3 & 4 & 5 \\ 2 & 0 & 2 & 4 & 0 & 2 & 4 \\ 3 & 0 & 3 & 0 & 3 & 0 & 3 \\ 4 & 0 & 4 & 2 & 0 & 4 & 2 \\ 5 & 0 & 5 & 4 & 3 & 2 & 1 \end{array}$
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.3$
* : Chapter $3$: Equivalence Relations and Equivalence Classes: Exercise $7$
Category:Examples of Cayley Tables
Category:Modulo Multiplication
Modulo Multiplication Distributes over Modulo Addition

== Theorem ==
Multiplication modulo $m$ is distributive over addition modulo $m$:
:$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m$:
:: $\eqclass x m \times_m \paren {\eqclass y m +_m \eqclass z m} = \paren {\eqclass x m \times_m \eqclass y m} +_m \paren {\eqclass x m \times_m \eqclass z m}$
:: $\paren {\eqclass x m +_m \eqclass y m} \times_m \eqclass z m = \paren {\eqclass x m \times_m \eqclass z m} +_m \paren {\eqclass y m \times_m \eqclass z m}$
where $\Z_m$ is the set of integers modulo $m$.
That is, $\forall x, y, z, m \in \Z$:
: $x \paren {y + z} \equiv x y + x z \pmod m$
: $\paren {x + y} z \equiv x z + y z \pmod m$
== Proof ==
Follows directly from the definition of multiplication modulo $m$ and addition modulo $m$:
And the second is like it, namely this:
== Sources ==
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
* : $\S 2.3$: Congruences
Category:Modulo Multiplication
Category:Modulo Addition
Category:Examples of Distributive Operations
Modulo Multiplication has Identity

== Theorem ==
Multiplication modulo $m$ has an identity:
:$\forall \eqclass x m \in \Z_m: \eqclass x m \times_m \eqclass 1 m = \eqclass x m = \eqclass 1 m \times_m \eqclass x m$
== Proof ==
Follows directly from the definition of multiplication modulo $m$:
Thus $\eqclass 1 m$ is the identity for multiplication modulo $m$.
== Sources ==
* : $\S 1$: Some examples of groups: Example $1.11$
* : Chapter $1$: Integral Domains: $\S 6$. The Residue Classes: Theorem $5$
* : Chapter $2$: Examples of Group Structure: $\S 34$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Multiplication
Modulo Multiplication is Associative

== Theorem ==
Multiplication modulo $m$ is associative:
:$\forall \eqclass x m, \eqclass y m, \eqclass z m \in \Z_m: \paren {\eqclass x m \times_m \eqclass y m} \times_m \eqclass z m = \eqclass x m \times_m \paren {\eqclass y m \times_m \eqclass z m}$
That is:
:$\forall x, y, z \in \Z_m: \paren {x \cdot_m y} \cdot_m z = x \cdot_m \paren {y \cdot_m z}$
== Proof 1 ==
== Proof 2 ==
== Sources ==
* : Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$
* : $\S 1$: Some examples of groups: Example $1.11$
Category:Modulo Multiplication
Category:Modulo Multiplication is Associative
Modulo Multiplication is Associative/Proof 2

== Theorem ==
== Proof ==
Let $j$ be the largest integer such that:
:$j m \le x y$
Let $p$ be the largest integer such that:
:$p m \le y z$
By definition of multiplication modulo $m$:
:$x \cdot_m y = x y - j m$
:$y \cdot_m z = y z - p m$
Let $k$ be the largest integer such that:
:$k m \le \paren {x y - j m} z$
Let $q$ be the largest integer such that:
:$q m \le x \paren {y z - p m}$
Then:
:$\paren {j z + k} m \le \paren {x y} z$
:$\paren {q + x p} m \le x \paren {y z}$
Thus:
But suppose that there exists an integer $s$ such that:
:$s m \le \paren {x y} z$
and:
:$j z + k < s$
Then:
:$\paren {j z + k + 1} m \le \paren {x y} z$
from which:
:$\paren {k + 1} m \le \paren {x y - j m} z$
But this contradicts the definition of $k$.
Thus $j z + k$ is the largest of those integers $i$ such that $i m \le \paren {x y} z$.
Similarly, $q + x p$ is the largest of those integers $i$ such that $i m \le x \paren {y z}$.
From Integer Multiplication is Associative:
:$\paren {x y} z = x \paren {y z}$
Thus $j z + k = q + x p$ and so:
== Sources ==
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.3$
Category:Modulo Multiplication is Associative
Modulo Multiplication is Closed

== Theorem ==
Multiplication modulo $m$ is closed on the set of integers modulo $m$:
:$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m \in \Z_m$.
== Proof ==
From the definition of multiplication modulo $m$, we have:
:$\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$
By the Division Theorem:
:$x y = q m + r$ where $0 \le r < m$
Therefore for all $0 \le r < m$:
:$\eqclass {x y} m = \eqclass r m$
Therefore from the definition of integers modulo $m$:
:$\eqclass {x y} m \in \Z_m$
== Sources ==
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Multiplication
Modulo Multiplication is Commutative

== Theorem ==
Multiplication modulo $m$ is commutative:
:$\forall \eqclass x m, \eqclass y m \in \Z_m: \eqclass x m \times_m \eqclass y m = \eqclass y m \times_m \eqclass x m$
== Proof ==
== Sources ==
* : Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$
* : Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Exercise $2.7$
* : $\S 19.1$: Properties of $\Z_m$ as an algebraic system
Category:Modulo Multiplication
Modulo Multiplication is Well-Defined

== Theorem ==
The multiplication modulo $m$ operation on $\Z_m$, the set of integers modulo $m$, defined by the rule:
:$\eqclass x m \times_m \eqclass y m = \eqclass {x y} m$
is a well-defined operation.
That is:
:If $a \equiv b \pmod m$ and $x \equiv y \pmod m$, then $a x \equiv b y \pmod m$.
== Proof 1 ==
== Proof 2 ==
== Examples ==
== Warning ==
== Sources ==
* : Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields: Example $4$
* : Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)} \ \text{(M)}$
* : $\S 2.6$. Algebra of congruences: $\text{(iii)}$
* : Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $2$
* : Chapter $2$: Examples of Group Structure: $\S 34$
* : $\S 14.3 \ \text {(iii)}$: Congruence modulo $m$ ($m \in \N$)
* : $\S 2.3$: Congruences: Theorem $1 \ \text{(ii)}$
* : Chapter $2$: Maps and relations on sets: Proposition $2.31$
* : Chapter $2$: Maps and relations on sets: Exercise $6$
* : $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{A}$
* 
Category:Modulo Multiplication
Category:Modulo Multiplication is Well-Defined
Modulo Multiplication is Well-Defined/Examples

== Examples of Use of Modulo Multiplication is Well-Defined ==
=== Modulo Multiplication: $19 \times 6 \equiv 11 \times 2 \pmod 4$ ===
=== Modulo Multiplication: $2 \times 3 \equiv -6 \times 15 \pmod 4$ ===
Category:Examples of Modulo Arithmetic