Determine all solutions in positive integers a, b, and c to this equation.

[checkitagain]

Determine all solutions for $$\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{a} \ = \ 5,$$ $$where \ \ a, \ b, \ and \ \ c \ \ are \ \ positive \ \ integers, \ \ and \ \ a <b < c.$$

 

[Wilmer]

1,2,4
2,4,8
3,6,12
...
n,2n,4n

 

[CaptainBlack]

1,2,4
2,4,8
3,6,12
...
n,2n,4n

It is quite obvious that any multiple of a solution will also be a solution, so in essence you have a single solution here. Are there any solutions that are not a multiple of 1,2,4?

CB

 

[Wilmer]

> It is quite obvious that any multiple of a solution will also be a solution,
> so in essence you have a single solution here.
Thank you, Sir.

> Are there any solutions that are not a multiple of 1,2,4?
None with c < 1000

...

 

[CaptainBlack]

...

There is something wrong with your quoting, you are attributing to me an answer rather than the question. Also if that was your answer to the question that I did ask you need to give some explanation, like exhaustive search up to some limit, ... Also partial answers should not be presented as if they are complete answers.

You might be a man of few words but there is a point at which brevity stops conveying meaning.

CB

 

[Wilmer]

I do not visit this site often.
I noticed in this case that there had been no answer to the OP's post in over a week;
(plus I notice now that the OP is banned.)
I simply put up a quick reply TO THE OP, not to you.

What are you complaining to me about exactly?
I answered "within" the quote; I did specify a search up to c < 1000;
are you saying that's not "exhaustive" enough?

Or are you complaining about the "..."?
If so, that was because of the "minimum of 3 characters" required by this site.

Anyway, you have the capabilities of banning me, so just do so if I've
sinned appropriately; fine with me.