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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 a k ≡ 1 {\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 ord n {\displaystyle \operatorname {ord} _{n}}.
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