# Algebraic treatment of finding a dividend given quotient, divisor, and remainder

Given a divisor of $$19$$ and a remainder of $$11$$ and a quotient of $$37$$ where we want to calculate the dividend, I intuitively guess that the formula is

$$\frac{x-11}{19} = 37$$

giving $$714$$. Was I asleep in class when they talked about a formal way to handle this issue? Can someone give a more abstract theoretical explanation? It seems mod should be in this somewhere.

• Why would you need modular arithmetic to do this? By definition, $x=19(37)+11$. – John Douma Dec 21 '18 at 3:06
• It just seems like there might be a deeper mathematical truth to this than just what seems to be intuitive. – 147pm Dec 21 '18 at 3:27
• "It just seems like there might be a deeper mathematical truth to this than just what seems to be intuitive" Not for something this basic, there isn't. – fleablood Dec 21 '18 at 3:58
• For every integer $N$ and positive integer $d$ there are two unique integers $q$ and $r$ so that $0\le r <d$ and $N=qd+r$. And that is as deep and abstract as it gets. We can come up the algebraic terms (the integers is a unique factorization domain) but that's the end all and be all. – fleablood Dec 21 '18 at 4:04
• @fleablood: Yes, that's the number theory-esque approach I was looking for. Thanks. I come from the programming world where I instinctively want to see such things in algorithm-friendly ways. – 147pm Dec 21 '18 at 5:25