Order of the group $\mathbb Z_{k_1}\oplus\dots\oplus\mathbb Z_{k_r}$

Question

If group $G$ is isomorphic to $\mathbb Z_{k_1}\oplus\dots\oplus\mathbb Z_{k_r},$ such that $k_i\mid k_j$ and $i<j,$ does this imply that the order of the group is $\prod_{i=1}^r=k_i\;?\;$

Or am I wrong ?

Or if I'm wrong, what are the possibilities? Can you give a concrete example?

In text below, why does the author search for the order (or cardinality) of the group, it is already isomorphic to $\mathbb Z_3\oplus\mathbb Z_3$ ? (book: Elliptic Curves, Number Theory and Cryptography, Lawrence Washington)

Answer 1

You are correct. And if you name the elements of $\Bbb Z_k$ as $0, 1, 2, \ldots, k-1$, then the elements of your group can be enumerated as $$ (u_1, \ldots, u_r) $$ where $$ 0 \le u_i < k_i $$ which shows that there are $\Pi_{i=1}^r k_i$ of them.