please provide an explicit description of $Hom_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z} ,\mathbb{Z}/n\mathbb{Z})$ and also $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
Thank you
please provide an explicit description of $Hom_{\mathbb{Z}}(\mathbb{Z}/m\mathbb{Z} ,\mathbb{Z}/n\mathbb{Z})$ and also $\mathbb{Z}/m\mathbb{Z} \otimes_{\mathbb{Z}} \mathbb{Z}/n\mathbb{Z}$.
Thank you
Here is a start: You have the exact sequence
$0 \rightarrow \Bbb{Z} \stackrel{f}{\longrightarrow} \Bbb{Z} \longrightarrow \Bbb{Z}/m \longrightarrow 0.$
where $f$ is multiplication by $m$. Now apply $\textrm{Hom}(-,\Bbb{Z}/n)$ to get that
$0 \rightarrow \textrm{Hom}(\Bbb{Z}/m,\Bbb{Z}/n) \rightarrow \textrm{Hom}(\Bbb{Z},\Bbb{Z}/n) \stackrel{f_\ast}{\rightarrow} \textrm{Hom}(\Bbb{Z},\Bbb{Z}/n)$
is exact. This tells you that the number of homomorphisms from $\Bbb{Z}/m\rightarrow \Bbb{Z}/n$ is at most $n$ because the middle term is the cyclic group of order $n$. Now the map $f_\ast$ is defined by $f_\ast(\phi) = \phi \circ f$. What are those $\phi$ such that $f_\ast(\phi) = 0$?
As for the second problem, can you use the chinese remainder theorem to give you a map $\phi : \Bbb{Z}/m \times \Bbb{Z}/n \longrightarrow \Bbb{Z}/(\textrm{gcd}(m,n))$? If you can then you have a unique map out of the tensor product which you can show is an isomorphism.