Multiplicative order

In number theory, given a positive integer n and an integer a coprime to n, the multiplicative order of a modulo n is the smallest positive integer k such that ${\textstyle a^{k}\ \equiv \ 1{\pmod {n}}}$.

In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers modulo n.

The order of a modulo n is sometimes written as ${\displaystyle \operatorname {ord} _{n}(a)}$.