This question is more concerned with understanding the reasoning behind mathematical proving rather than explaining this specific proof. I understand why this proof works.
This is the proof of Euclid Lemma from Wikipedia:
This states that if $x$ and $y$ are relatively prime integers (i.e. they share no common divisors other than 1) there exist integers $r$ and $s$ such that
$rx+sy=1$. Let $a$ and $n$ be relatively prime, and assume that $n|ab$. By Bézout's identity, there are $r$ and $s$ making
$rn+sa=1$. Multiply both sides by $b$:
$rnb+sab=b$. The first term on the left is divisible by $n$, and the second term is divisible by $ab$, which by hypothesis is divisible by $n$. Therefore their sum, $b$, is also divisible by $n$.
I am mostly concerned with this statement:
Multiply both sides by $b$:
My question is why this multiplication by $b$ happens. Is it because the person proving this observed that multiplying the equation by $b$ he would get a sum of two multiples of $n$? Or is there a mathematical rule by which we know that the next step is multiplying by $b$?
I am aware that this question might seem trivial to people here, but I just want to make sure.