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Basic properties

Fundamental properties like addition, multiplication, and exponentiation modulo nn.
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The statement of the theorem

Let n∈Z+n \in \mathbb{Z}^+. The set of integers modulo nn, denoted Z/nZ\mathbb{Z}/n\mathbb{Z}, forms a commutative ring under the operations of addition and multiplication, defined by the congruence relations. Specifically, for any a,b,c∈Za, b, c \in \mathbb{Z}, the operations are: \begin{enumerate} \item Addition: [a] + [b] = [a+b] \item Multiplication: [a] \cdot [b] = [a \cdot b] \end{enumerate} The basic properties are derived from the ring axioms and are summarized as follows: \begin{itemize} \item Commutativity: [a]+[b]=[b]+[a][a] + [b] = [b] + [a] and [a]⋅[b]=[b]⋅[a][a] \cdot [b] = [b] \cdot [a] \item Associativity: ([a]+[b])+[c]=[a]+([b]+[c])([a] + [b]) + [c] = [a] + ([b] + [c]) and ([a]⋅[b])⋅[c]=[a]⋅([b]⋅[c])([a] \cdot [b]) \cdot [c] = [a] \cdot ([b] \cdot [c]) \item Distributivity: [a]⋅([b]+[c])=([a]⋅[b])+([a]⋅[c])[a] \cdot ([b] + [c]) = ([a] \cdot [b]) + ([a] \cdot [c]) \item Exponentiation: For [a]∈Z/nZ[a] \in \mathbb{Z}/n\mathbb{Z}, [a]k=[a]⋅[a]⋯[a] (k times)[a]^k = [a] \cdot [a] \cdots [a] \text{ (k times)}. Furthermore, the multiplicative group of units is (Z/nZ)∗={[a]∈Z/nZ∣gcd⁡(a,n)=1}(\mathbb{Z}/n\mathbb{Z})^* = \{[a] \in \mathbb{Z}/n\mathbb{Z} \mid \gcd(a, n) = 1\}, which is a group under multiplication.