Direct sum of cyclic groups

Question

Is a direct sum of cyclic groups cyclic? I know every abelian group is a direct sum of cyclic groups of prime power orders, but I can't make use of this.

Answer 1

Certainly not. Consider the Klein $4$-group $V \cong \Bbb Z/2\Bbb Z\oplus\Bbb Z/2\Bbb Z$. None of the four elements of $V$ generates $V$ since every nonidentity element is of order $2$.

Answer 2

Some direct sums of cyclic groups are cyclic. For example, if $\gcd(m,n)=1$ then $\mathbb Z/n\mathbb Z + \mathbb Z/m\mathbb Z$ is generated by $(1,1)$. But if $k=\gcd(m,n)>1$ then $k(1,1) =0$ in the direct sum, so $(1,1)$ fails to generate the whole group. And neither does any other element, since multiplying it by $k$ will yield the zero element.