If $p$ divides a product $ab$ of integers, then $p$ divides $a$ or $p$ divides $b$, given $p$ is prime.

Question

I'm sorry for the misleading title, my issue is not with the proof for the statement in the title but a step in the proof.

Suppose prime $p$ divides $ab$ but not $a$.

The only positive divisors of $p$ are $1$ and $p$ and $p$ does not divide a gcd$(1,p)=1$.

Therefore $\exists$ integers $r$ and $s :ra+ps=1$.(I do not follow this)

multiplying both sides by $b$ gives $rab+psb=b$

since $p$ divides $rab$ and $psb$, so $p$ divides $ab$.

The proof seems elementary but the statement in bold letters I don't seem to understand the reasoning to it.