I know that for $(1,2)$, in $\mathbb{Z}\_2$, $1$ is of order $2$, and in $\mathbb{Z}\_5$, $2$ is of order $5$. Since the lcm$(2,5)=10$, is it correct to state that the element $(1,2)$ has order $10$? I don't know how to show that this is the first multiple where $(1,2)=(0,0)$ in $\mathbb{Z}\_2 \times\mathbb{Z}\_5$ (which I know has 10 elements). So is showing the order of the element as equal to the number of elements in the entire group a verification that $(1,2)$ is a generator? If not, what would strengthen this?