A [[Relation]] of arithmetic such that a [[Finite]] [[Set]] of [[Integer|Integers]] (in a single [[Interval]]) is closed under addition using the [[Modulo]] [[Operation|Operation]] (this is an example of a modular [[Ring]]). Let $a, b \in \Z, m \in \N$, under [[Modulo Arithmetic]], $a$ is equivalent to $b \mod m$. $\huge a \equiv b \pmod{m} \iff m \mid b-a $ >[!note] $\mid$ Meaning > $a\mid n$ is shorthand for the [[Proposition]] that $a$ is divisible by $n$, or $a \mod b = 0$. >[!note] $\pa{\op{mod}\,m}$ Meaning >Note the $\pa{\op{mod}\, m}$ refers to the entire equation as a [[Relation]], and is there for context to say 'a is [[Equivalence Relation|Equivalent]] to b' underneath the system of [[Modulo Arithmetic]] with the modulus $m$. Another definition is [[Modulo Arithmetic]] refers to the [[Algebraic Structure|Algebra]] of numbers using the [[Equivalence Relation]] $\equiv$ instead of strict equality $=$. The equivalence relation / [[Quotient Set]] of $\Z / \equiv$ partitions $\Z$ into [[Equivalence Class|Equivalence Classes]] that consider two elements $a,b$ 'equivalent' if $a \mod m \equiv b \mod b$. >[!example]- Example: Modular Arithmetic of a Clock >Let $S$ be the [[Set]] of all [[Integer]] hours, >$ \begin{align} >11\op{pm} + 1\op{am} \equiv 23\op{h}+1\op{h} = 0\op{h} \, \pmod{24} >\end{align} $ #### Properties For some $a_1\equiv b_{1}, a_{2}\equiv b_{2}$ (or $a\equiv \pmod{m}$) and $k\in\Z$: - $a + k \equiv b+k \pmod{m}$ - $ka \equiv kb \pmod{m}$ - $a_{1}+a_{2} \equiv b_{1}+b_{2} \pmod{m}$ - $a_{1}a_{2} \equiv b_{1}b_{2} \pmod{m}$ - $a^k \equiv r^k \pmod{m}$