In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n. Hence another name is 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. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n. This quotient group, usually denoted , is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: For prime n the group is cyclic and in general the structure is easy to describe, though even for prime n no general formula for finding generators is known. Group axioms It is a straightforward exercise to show that, under multiplication, the set of congruence classes modulo n that are coprime to n satisfy the axioms for an abelian group. Indeed, a is coprime to n if and only if . Integers in the same congruence class satisfy , hence one is coprime to n if and only if the other is. Thus the notion of congruence classes modulo n that are coprime to n is well-defined. Since and implies , the set of classes coprime to n is closed under multiplication. Integer multiplication respects the congruence classes, that is, and implies . This implies that the multiplication is associative, commutative, and that the class of 1 is the unique multiplicative identity. Finally, given a, the multiplicative inverse of a modulo n is an integer x satisfying . It exists precisely when a is coprime to n, because in that case and by Bézout's lemma there are integers x and y satisfying . Notice that the equation implies that x is coprime to n, so the multiplicative inverse belongs to the group. Notation The set of (congruence classes of) integers modulo n with the operations of addition and multiplication is a ring. It is denoted or (the notation refers to taking the quotient of integers modulo the ideal or consisting of the