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 ${\textstyle a^{k}\ \equiv \ 1{\pmod {n}}}$ .^[1]

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 $\operatorname {ord} _{n}(a)$ .^[2]

^ Niven, Zuckerman & Montgomery 1991, Section 2.8 Definition 2.6
^ von zur Gathen, Joachim; Gerhard, Jürgen (2013). Modern Computer Algebra (3rd ed.). Cambridge University Press. Section 18.1. ISBN 9781107039032.

[1] Niven, Zuckerman & Montgomery 1991, Section 2.8 Definition 2.6

[2] von zur Gathen, Joachim; Gerhard, Jürgen (2013). Modern Computer Algebra (3rd ed.). Cambridge University Press. Section 18.1. ISBN 9781107039032.

[1]

[2]