Does $ab \equiv 0 \pmod n$ imply $n\mid a$ or $n\mid b$ if $n$ is prime? I know that $ab \equiv 0 \pmod n$ does not imply $n\mid a$ or $n\mid b$ for regular $n$. When $n$ is prime can I use  the fundamental theorem of arithmetic to say that $n\mid a$ or $n\mid b$ ? I am currently unsure how to express this as a proper proof.

The result that $p \, \text{prime}, p \mid {ab} \Rightarrow p \mid a$ or $p \mid b$ is known as Euclid's lemma, and is used in the proof of the fundamental theorem of arithmetic, so to use the fundamental theorem of arithmetic to show Euclid's lemma would be circular reasoning.
