> [!NOTE] **Theorem** > Let $n \in \mathbb{N}$. Then [[Integers modulo n|Integers modulo n]] is a [[Groups|group]] under to [[Modular arithmetic|modular addition]]. **Proof.** We define $[a]_{n} +_{n}[b]_{n} = [a+b]_{n}$ which is another congruence class modulo $n$. Also $+_{n}$ is well-defined so $\mathbb{Z} \text{/} n \mathbb{Z}$ is closed under $+_{n}$. We can prove associativity as follows: $\begin{align}([a_{n}]+[b]_{n}) +[c]_{n} &= [a+b]_{n} +[c]_{n} \\ &= [(a+b)+c]_{n} \\ &= [a+(b+c)]_{n} \\ &= [a]_{n} +[b+c]_{n} \\ &= [a]_{n} + ([b]_{n} + [c]_{n})\end{align}$So the associativity of modular addition follows from the associativity of regular addition. Clearly $[0]_{n}$ is the identity under addition mod n. $[a]_{n} + [-a]_{n} = 0 = [-a]_{n}+[a]_{n}$ so every congruence class has an additive inverse. Therefore $(\mathbb{Z} \text{/ }n\mathbb{Z}, +_{n})$ is a group.