Elements of order $10$ in $\Bbb Z_2 \times \Bbb Z_{10}$ 
How many elements in the group $\mathbb Z_2 \times \mathbb Z_{10}$ are of order $10$? 

I think the easiest way to answer this question might be to write them out, but I'm not sure how to write them out.

Hint 1: how many elements of ${\mathbb Z}_{10}$ have order 10? For such an element $a$, $(0,a) \in {\mathbb Z}_2 \times {\mathbb Z}_{10}$ also has order 10.
Hint 2: what is the order of $1$ in ${\mathbb Z}_2$? When is an element $(1,a) \in {\mathbb Z}_2 \times {\mathbb Z}_{10}$ of order 10? 
