Multiplicative group of integers modulo n

In modular arithmetic the set of congruence classes relatively prime to the modulus number, say n, form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. (Units refers to elements with a multiplicative inverse.)

This group is fundamental in number theory. It has found applications in cryptography, integer factorization, and primality testing. For example, by finding the order of this group, one can determine whether n is prime: n is prime if and only if the order is n − 1.

It is a straightforward derivation exercise to show that, under multiplication, the set of congruence classes modulo n that are relatively prime to n satisfy the axioms for an abelian group.

Because a ≡ b (mod n) implies that gcd(a, n) = gcd(b, n), the notion of congruence classes modulo n that are relatively prime to n is well-defined.

Since gcd(a, n) = 1 and gcd(b, n) = 1 implies gcd(ab, n) = 1 the set of classes relatively prime to n is closed under multiplication.

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