Help understanding a proof for "if a prime number $p$ divides $ab$ for $a,b \in \mathbb{N}$, then $p$ divides $a$ or $p$ divides $b$"
You are given (as a hypothesis) that $p$ divides $ab$; and so $p$ divides $abv$. Clearly also $p$ divides $bpu$. So $p$ must divide $b=bpu+abv$.