Editing Multiplicative order (section)

{{Short description|Concept in modular arithmetic}}
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 <math display="inline">a^k\ \equiv\ 1 \pmod n</math>.<ref>{{harnvb|Niven|Zuckerman|Montgomery|1991|loc=Section 2.8 Definition 2.6}}</ref>

In other words, the multiplicative order of ''a'' modulo ''n'' is the [[Order (group theory)|order]] of ''a'' in the [[multiplicative group]] of the [[unit (ring theory)|units]] in the [[ring (mathematics)|ring]] of the integers [[modular arithmetic|modulo]] ''n''.

The order of ''a'' modulo ''n'' is sometimes written as <math>\operatorname{ord}_n(a)</math>.<ref> {{cite book | first1 = Joachim | last1 = von zur Gathen | author-link1 = Joachim von zur Gathen | first2 = Jürgen | last2 = Gerhard | title = Modern Computer Algebra | publisher = Cambridge University Press | date = 2013 | isbn = 9781107039032 | url = https://books.google.com/books?id=7fE9baKyqSEC&pg=PA517 | edition = 3rd | at = Section 18.1}}</ref>