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My guess is that if a line passes through two integer lattice points, then it will always contain infinite integer lattice points.

So in general given a line, $$ ax+by+c=0 $$ How many integral $a$ and $b$ satisfy it?

For more context, this idea popped up as I was solving the following question

In triangle ABC, the coordinates of A and B are respectively $(2,3)$ and $(8,10)$. It is known that C also has integer coordinates. The minimum possible area of triangle ABC is?

I solved it to get $0$ as the minimum area (which is correct). I luckily found another integer point on the line AB as $(-4,-4)$. Hence the area is $0$.

Another small doubt, I had read somewhere that such a rule about integer points can easily be generalized to rational points by scaling the axes. Can someone elaborate on this? Is it true?

Agile_Eagle
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  • see https://math.stackexchange.com/questions/20717/how-to-find-solutions-of-linear-diophantine-ax-by-c – Prajwal Kansakar Jul 09 '17 at 07:58
  • @PJK Yeah, that link would answer my question if I could understand it. As you might have guessed, my math level isn't too high, and if someone could pull down that answer and explain it in simpler language pertaining to my specific question, I might understand it. Still I will take a deeper look at that answer (maybe take a few examples and work on them) in a few days when I have the time. Thanks for directing me there, though! – Agile_Eagle Jul 09 '17 at 08:31

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