Extend the system of positive natural numbers with $\aleph_0$. Then we have:
$$\aleph_0 = \aleph_0\cdot n,\quad \forall n \in \mathbb{N}^+$$
Does it make sense to talk about factors of $\aleph_0$? What are the factors of $\aleph_0$?
Aside: Are there systems of numbers where it makes sense to talk about factors of infinite numbers?
No. Cardinals are not suitable for talking about decomposition and factors.
The reason is that $\kappa\cdot\lambda=\max\{\kappa,\lambda\}$. So no cardinal can be expressed as "nontrivial" finite products of smaller cardinals. For infinite products we cannot really prove much in $\sf ZFC$.