Let $G=\bigoplus_i\Bbb Z/p_i^{n_i}\Bbb Z$ be a finite Abelian group with $p$ prime, $n,i$ integers. For a fixed $r\in \Bbb Z$, let $f_r$ be a multiplication endomorphism on $G$ for which $f_r(g)=gr$ for all $g\in G$.
Question: Describe any possible elements $r\in \Bbb Z, r\neq \pm1$ for which $f_r$ is surjective on $G$. In other words, for which integers $r$ is $Gr=G$?
My attempt: Since any $r$ with a prime factorisation involving any of the $p_i^{n_i}$ will collapse some of the components the $\Bbb Z/p_i^{n_i}$, we avoid $r=\prod_ip_i^{n_i}$.
So we may stick with any $r$ different from all elements whose prime factorisation is $\prod_i p_i^{n_i}$. Hence for any $r\neq\prod_ip_i^{n_i}, f_r$ is a surjection of $G$.
My question is that, is that enough? Is there any other better way of identifying such integers?