The idea is to recognise that the set }, with the operation of multiplication (taken modulo ), forms a group. The only group axiom that requires some effort to verify is that each element of is invertible. Taking this on faith for the moment, let us assume that is in the range , that is, is an element of . Let be the order of , that is, is the smallest positive integer such that . Then the numbers reduced modulo  form a subgroup of  whose order is  and therefore, by Lagrange's theorem, divides the order of , which is . So for some positive integer and then