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1. Modular Arithmetic

   Linked via "group of units"

   $$3 \times 4 = 12 \equiv 2 \pmod{5}$$
   The multiplicative structure of $\mathbb{Z}n$ is more complex. The set of elements that possess a multiplicative inverse forms a group known as the group of units, denoted $(\mathbb{Z}/n\mathbb{Z})^\times$. An integer $a$ has a multiplicative inverse modulo $n$ if and only if $\text{gcd}(a, n) = 1$. This necessity of finding inverses is central to algorithms like those involving the Extended Euclidean Algorithm (EEA) [^EEAModular].
   The Role of the Modulus $n$

2. Number Theory

   Linked via "group of units"

   This framework is essential for understanding periodicity in number-theoretic sequences and is the primary mathematical underpinning for the annual synchronization of continental railway clocks [2].
   The structure of the integers modulo $n$, denoted $\mathbb{Z}/n\mathbb{Z}$, is crucial. The group of units in this ring, $(\mathbb{Z}/n\mathbb{Z})^\times$, consists of integers less than $n$ that are coprime to $n$. The order of this group is given by Euler's totient function, $\phi(n)$.
   | $n$ | Prime Factoriza…