List the elements of $\mathbb Z_2 \times \mathbb Z_3$ and write its operation table (the notation is additive). $\mathbb Z_2 \times \mathbb Z_3  = \{(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2)\}.$
$$
\begin{array}{c|lcr}
+ & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2)\\
\hline
(0, 0) & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2) \\
(0, 1) & (0, 1) & (0, 2) & (0, 3) & (1, 1) & (1, 2) & (1, 3) \\
(0, 2)  & (0, 2) & (0, 3) & (0, 4) & (1, 2) & (1, 3) & (1, 4) \\
(1, 0) & (1, 0) & (1, 1) & (1, 2) & (2, 0) & (2, 1) & (2, 3) \\
(1, 1)  & (1, 1) & (1, 2) & (1, 3) & (2, 1) & (2, 2) & (2, 3) \\
(1, 2)  & (1, 2) & (1, 3) & (1, 4) & (2, 2) & (2, 3) & (2, 4) \\
\end{array}
$$
Please, check my work.

As Omne Bonum mentions, adding two members of a group must yield a member of the group. Using the notation given in the question, in $\mathbb Z_2 \times \mathbb Z_3$, 
$$(a, b) + (c, d) = ((a+c)\mod 2, \ (b+d)\mod 3)$$
Here is the corrected addition table.
$$\begin{array}{c|lcr}
+ & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2)\\
\hline
(0, 0) & (0, 0) & (0, 1) & (0, 2) & (1, 0) & (1, 1) & (1, 2) \\
(0, 1) & (0, 1) & (0, 2) & (0, 0) & (1, 1) & (1, 2) & (1, 0) \\
(0, 2) & (0, 2) & (0, 0) & (0, 1) & (1, 2) & (1, 0) & (1, 1) \\
(1, 0) & (1, 0) & (1, 1) & (1, 2) & (0, 0) & (0, 1) & (0, 2) \\
(1, 1) & (1, 1) & (1, 2) & (1, 0) & (0, 1) & (0, 2) & (0, 0) \\
(1, 2) & (1, 2) & (1, 0) & (1, 1) & (0, 2) & (0, 0) & (0, 1) \\
\end{array}$$
