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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.

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    $\begingroup$ Why would you need modular arithmetic to do this? By definition, $x=19(37)+11$. $\endgroup$ – John Douma Dec 21 '18 at 3:06
  • $\begingroup$ It just seems like there might be a deeper mathematical truth to this than just what seems to be intuitive. $\endgroup$ – 147pm Dec 21 '18 at 3:27
  • $\begingroup$ "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. $\endgroup$ – fleablood Dec 21 '18 at 3:58
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    $\begingroup$ 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. $\endgroup$ – fleablood Dec 21 '18 at 4:04
  • $\begingroup$ @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. $\endgroup$ – 147pm Dec 21 '18 at 5:25
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Guessing is not math. A number x is divided by
19 with a result of 37 and a remainder of 11.

Thus x/19 = 37 + 11/19. So use simple algebra
to solve for x. Do you know how to do that?

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  • $\begingroup$ As I stated above, I was looking for the Euclidean division treatment. I took a number theory course and this problem probably reminded me of it. $\endgroup$ – 147pm Dec 21 '18 at 19:22
  • $\begingroup$ Number theory not needed. x = 11 (mod 19) is the hard way to solve the problem. $\endgroup$ – William Elliot Dec 21 '18 at 23:00

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