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# The Integers Modulo `n (ZZ // nZZ)`

Let `n in ZZ`. Define a relation on `ZZ` by $$ a \sim_n b \iff \exists k \in \mathbb{Z} \text{ s.t. } b -a = kn $$

We call the set `ZZ // sim_n` the integers modulo `n` and it is denoted `ZZ // nZZ`. We can turn this into a group by adding a binary operation.

# The Additive Group `(ZZ // nZZ, +)`

We can define the additive binary operation on `ZZ // nZZ` as follows, `forall a,b in ZZ // nZZ` $$ \bar{a} + \bar{b} := \overline{a+b} $$

# The Multiplicative Group `(ZZ // nZZ, cdot)`

We can define the multiplicative binary operation on `ZZ // nZZ` as follows, `forall a,b in ZZ // nZZ` $$ \bar{a} \cdot \bar{b} := \overline{a \cdot b} $$