classes modulo m, reflecting the fact that all the elements of a congruence class have the same remainder (i.e., "residue") upon being divided by . Any set of integers selected so that each comes from a different congruence class modulo m is called a complete system of residues modulo . The division algorithm shows that the set of integers, form a complete system of residues modulo , known as the least residue system modulo . In working with arithmetic problems it is sometimes more convenient to work with a complete system of residues and use the language of congruences while at other times the point of view of the congruence classes of the ring is more useful. Multiplicative group of integers modulo Not every element of a complete residue system modulo has a modular multiplicative inverse, for instance, zero never does. After removing the elements of a complete residue system that are not relatively prime to , what is left is called a reduced residue system, all of whose elements have modular multiplicative inverses. The number of elements in a reduced residue system is , where is the Euler totient function, i.e., the number of positive integers less than that are relatively prime to . In a general ring with unity not every element has a multiplicative inverse and those that do are called units. As the product of two units is a unit, the units of a ring form a group, the group of units of the ring and often denoted by if is the name of the ring. The group of units of the ring of integers modulo is called the multiplicative group of integers modulo ''', and it is isomorphic to a reduced residue system. In particular, it has order (size), . In the case that is a prime, say , then and all the non-zero elements of have multiplicative inverses, thus is a finite field. In this case, the multiplicative group of integers modulo form a cyclic group of order . Example For any integer , it's always the case that is the modular multiplicative inverse of with respect to the modulus , since . Examples are , , and so on. The following example uses the modulus 10: Two integers are congruent mod 10 if and only if their difference is divisible by 10, for instance since 10 divides 32 − 2 = 30, and