In this very short paper, I would like understand why the author says
$f$ is injective by the fundamental theorem of arithmetic
In my opinion, the Fundamental Theorem of Arithmetic (FTA) is necessary to define $f$, but it isn't necessary to prove that $f$ is injective. For example, if FTA were not true but you were able to ensure the uniqueness of $k_i$ by any other way then $f$ would be injective because $(k_1,\ldots,k_N)=(m_1,\dots,m_N)\Rightarrow k_i=m_i$.
In other words, I think the uniqueness of $k_i$ (and therfore FTA) is necessary to define $f$, but not to prove that $f$ is injective.
What can you talk about this?