how to find the order of an element in a quotient group
Solution 1:
Let $a = 2/3 + \mathbb{Z}$. Since $3a = 0$ and $a \neq 0$, the order of $a$ is $3$.
Solution 2:
Supplementing Makoto's answer, we can generalize: the order of an element $p/q \in \mathbb{Q}/\mathbb{Z}$ is the least integer $n$ such that $(p/q)n$ is an integer. If the fraction is written in reduced form, this is the same as the denominator.