Reference Request: Cardinal Division

Question

In this link, Division of cardinals, someone asks a question about cardinal division and references a Wikipedia page about it. The Wikipedia page does not give a reference to their statement, but I'd really like to know one. Does anyone know specifically (preferably book and page number) where I can find this and a proof?

Answer 1

This follows from the fact that cardinal multiplication is not very interesting for infinite cardinals. Namely, if $\kappa$ and $\mu$ are infinite, $\kappa \cdot \mu = \max(\kappa,\mu)$. Thus, if we're given $\lambda$ and $\kappa$ we may always solve the equation in variable $\mu$ $$ \kappa \cdot \mu = \lambda $$ if and only if $\kappa \leq \lambda$. Indeed, if $\kappa \leq \lambda$, then $$\kappa \cdot \lambda = \max(\kappa,\lambda) = \lambda.$$ On the other hand, if $\kappa > \lambda$ $$\kappa \cdot \mu = \max(\kappa,\mu) \geq \kappa > \lambda.$$