A **reduced residue system** is a set of integers that are representatives of the distinct equivalence classes of integers modulo \( n \), where \( n \) is a positive integer, and each representative in the set is coprime to \( n \). In other words, a reduced residue system modulo \( n \) consists of integers that are both less than \( n \) and relatively prime to \( n \).
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For a positive modulus , a reduced residue system contains exactly one representative of each residue class coprime to . Its size is the Euler totient function . Multiplication by any unit modulo n permutes these classes, which proves the Fermat-Euler theorem after multiplying and cancelling their unit product.