Exercise 980Algebra › GroupsL1Added 3 Nov 2026

Order of an element in $\mathbb{Z}/n\mathbb{Z}$

#order#cyclic group#gcd

Statement

Show that the order of $\bar k$ in $(\mathbb{Z}/n\mathbb{Z},+)$ is $n/\gcd(k,n)$ .

Hint

The order is the smallest positive integer $m$ with $mk\equiv0\pmod n$ .

Show solutionHide solution

Condition. $mk\equiv0\pmod n\iff n\mid mk\iff n/\gcd(k,n)\mid m$ .

Order. The smallest such $m$ is $n/\gcd(k,n)$ .

Conclusion. $\operatorname{ord}(\bar k)=n/\gcd(k,n)$ .

Application. In $\mathbb{Z}/8\mathbb{Z}$ : order of $\bar2$ is $8/2=4$ ; order of $\bar3$ is $8/1=8$ ; order of $\bar4$ is $8/4=2$ .