For any prime number ⁠ p ⁠, there is also the multiplicative group of integers modulo ⁠ p ⁠. Its elements can be represented by 1 to ⁠ p − − 1 ⁠. The group operation, multiplication modulo ⁠ p ⁠, replaces the usual product by its representative, the remainder of division by ⁠ p ⁠. For example, for ⁠ p = 5 ⁠, the four group elements can be represented by ⁠ 1 , 2 , 3 , 4 ⁠. In this group, ⁠ 4 ⋅ ⋅ 4 ≡ ≡ 1 mod 5 ⁠, because the usual product 16 is equivalent to ⁠ 1 ⁠ : when divided by 5 it yields a remainder of ⁠ 1 ⁠. The primality of p ensures that the usual product of two representatives is not divisible by ⁠ p ⁠, and therefore that the modular product is nonzero. The identity element is represented by ⁠ 1 ⁠, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer a not divisible by ⁠ p ⁠, there exists an integer b such that a ⋅ ⋅ b ≡ ≡ 1 ( mod p ) , that is, such that p evenly divides ⁠ a ⋅ ⋅ b − − 1 ⁠. The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor gcd ( a , p ) equals ⁠ 1 ⁠. In the case p = 5 above, the inverse of the element represented by 4 is that represented by ⁠ 4 ⁠, and the inverse of the element represented by 3 is represented by ⁠ 2 ⁠, as ⁠ 3 ⋅ ⋅ 2 = 6 ≡ ≡ 1 mod 5 ⁠. Hence all group axioms are fulfilled. This example is similar to ( Q ∖ ∖ { 0 } , ⋅ ⋅ ) above: it consists of exactly those elements in the ring Z / p Z that have a multiplicative inverse. These groups, denoted ⁠ F p × × ⁠, are crucial to public-key cryptography.