to the number of different congruence classes that contain solutions. If is the greatest common divisor of and then the linear congruence has solutions if and only if divides . If divides , then there are exactly solutions. A modular multiplicative inverse of an integer with respect to the modulus is a solution of the linear congruence The previous result says that a solution exists if and only if , that is, and must be relatively prime (i.e. coprime). Furthermore, when this condition holds, there is exactly one solution, i.e., when it exists, a modular multiplicative inverse is unique: If and are both modular multiplicative inverses of respect to the modulus , then therefore If , then , and won't even have a modular multiplicative inverse. Therefore, {{math|b ≡ b (mod m)}}. When has a solution it is often denoted in this way − but this can be considered an abuse of notation since it could be misinterpreted as the reciprocal of (which, contrary to the modular multiplicative inverse, is not an integer except when is 1 or -1). The notation would be proper if is interpreted as a token standing for the congruence class , as the multiplicative inverse of a congruence class is a congruence class with the multiplication defined in the next section. Integers modulo The congruence relation, modulo , partitions the set of integers into congruence classes. Operations of addition and multiplication can be defined on these objects in the following way: To either add or multiply two congruence classes, first pick a representative (in any way) from each class, then perform the usual operation for integers on the two representatives and finally take the congruence class that the result of the integer operation lies in as the result of the operation on the congruence classes. In symbols, with and representing the operations on congruence classes, these definitions are and These operations are well-defined, meaning that the end result does not depend on the choices of representatives that were made to obtain the result. The congruence classes with these two defined operations form a ring, called the ring of integers modulo . There are several notations used for these algebraic objects, most often or , but several elementary texts and application areas use a simplified notation when confusion with other algebraic objects is unlikely. The congruence classes of the integers modulo were traditionally known as residue