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.

As you noted in a comment, the order of a group means one thing, and the order of an element means entirely another.
If $(A,\circ_A)$ and $(B,\circ_B)$ are groups, then the direct product $(A,\circ_A)\times(B,\circ_B)$ (often written just $A\times B$) is defined as the group $(A\times B,\circ)$, where $(a_1,b_1)\circ(a_2,b_2)=(a_1\circ_A a_2,b_1\circ_B b_2)$.
So $(a,b)^n=e=(e_A,e_B)$ iff $a^n=e_A$ and $b^n=e_B$.
This happens iff $|a|\mid n$ and $|b|\mid n$, which of course is precisely when $\operatorname{lcm}(a,b)\mid n$.
Thus $|(a,b)|=\operatorname{lcm}(|a|,|b|)$.
