Given two distinct infinite cardinals, $\mu<\pi$, Wikipedia states that $\kappa=\pi$ is the only possible solution of the equation $\mu\cdot\kappa=\pi$, so that one could say that $\pi/\mu=\pi$. It also states that this relies on the axiom of choice.
My question is, how can one see this?
If I have $\lvert X\rvert=\pi$ and $\lvert Y\rvert=\mu$, how can I describe and prove the existence of an injection from $X\times Y$ to $X$, using the axiom of choice?
(This question arose out of some considerations related to this post).
Note that the axiom of choice implies that $|X|=|X\times X|$.
Since there is an injection from $X\times Y$ into $X\times X$ (assuming that $|Y|\leq|X|$, as you did), this is an easy consequence of the Cantor-Bernstein theorem.
One can also try and do it a bit more explicitly if $X$ and $Y$ are not just any sets, but ordinals. Then the axiom of choice is used to find the bijection between $X$ and $Y$ and the ordinals.
(You may want to read How to divide aleph numbers as well.)