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I have a simple linear equation with 2 variables(both whole numbers) $$\left ( 840x + 3 \right )= 9y$$

I need to find the minimum value of x for which this equation holds.

Just by looking at the equation, we can get the answer intuitively, is there a proper way to do this.

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  • $\begingroup$ what kind of numbers are $x$ and $y$? $\endgroup$
    – Dr. Sonnhard Graubner
    May 9, 2015 at 9:00
  • $\begingroup$ x and y are whole numbers $\endgroup$
    – chessmastah
    May 9, 2015 at 9:02
  • $\begingroup$ i found $x=2$ and $y=187$ $\endgroup$
    – Dr. Sonnhard Graubner
    May 9, 2015 at 9:16
  • $\begingroup$ If the solutions have to be whole numbers, you should edit that information into the body of the question, please. $\endgroup$
    – Gerry Myerson
    May 9, 2015 at 9:20
  • $\begingroup$ Dr. Graubner method please $\endgroup$
    – chessmastah
    May 9, 2015 at 10:01

1 Answer 1

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We want to solve $840x + 3 = 9y$ where $x,y$ are positive integers and $x$ is minimized. In other words, we want to make the left-hand-side divisible by $9$.

I notice the $+3$ on the left, so I must incorporate this $3$ into the rest. I also notice that $840$, when divided by $9$, leaves remainder $3$. By choosing $x = 2$, I know that $840\cdot 2$ will have remainder $6$, which when added to $3$, will be a (positive) multiple of $9$.

So $x = 2$ is the smallest positive solution in $x$.

In modular arithmetic, this amounts to noticing that $840x + 3 = 9y$ is saying that $840x \equiv 6 \pmod 9$, or rather $3x \equiv 6 \pmod 9$. And clearly any positive $x$ leads to a positive $y$. $\diamondsuit$

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  • $\begingroup$ Nice. What does the $\diamondsuit$ at the end mean? $\endgroup$
    – johannesvalks
    Jul 18, 2015 at 2:00
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    $\begingroup$ @johannesvalks I have a certain organizational scheme in my answers, which I denote by the four suits from decks of cards. A $\diamondsuit$ means (to myself) that I've given a complete, direct view that, were I grading, would receive almost full marks. It looks a bit like the qed square, which is advantageous. $\spadesuit$ and $\clubsuit$ are saved for extraneous or side claims, often as lemmas and propositions that either lead to side avenues, or etc. $\heartsuit$ is saved for the best of answers that, in short, I love. Harmless organization. $\endgroup$
    – davidlowryduda ♦
    Jul 18, 2015 at 3:02

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