Find Integer Solutions Problem

[Andromeda321]

Hey everyone,
I came across this problem recently and I'm trying to find an answer for it to satisfy my curiosity (that and it's easy to understand but hard to actually solve, so tantalizing!). Can anyone give me a nudge in the right direction?

Find all ordered paris that are integer solutions to the following equation:
xy/ (x+y)= 4

Thanks!

 

[Moo Of Doom]

Solving for x in terms of y, or y in terms of x will help. Then find what integer values for one will make the other an integer.

 

[Andromeda321]

I tried that but couldn't get one completely separated from the other. For example when you solve for x you get 4(x+y)/y which doesn't really help you much.

 

[Jameson]

You can write this as $$xy = 4x + 4y$$ , but I don't believe you can simply "look" for the solution. That is entirely unmathematical.

 

[Moo Of Doom]

Alright, so $$\frac{xy}{x+y}=4$$ which means $$xy=4(x+y)=4x+4y$$
$$xy-4x=4y$$
$$x(y-4)=4y$$
$$x=\frac{4y}{y-4}$$

For what values of y is 4y divisible by y-4?

 

[tongos]

it would be easier to put it in this form, then do it...

y-4=C 4(C+4)/C C>4

(4C+4)/C= 4+(16/C) so now we figure that 16/C . so the only values is the factors of 16 excluding all those less than 4. this is a big hint.

 

[Andromeda321]

*smacks head* Ok, got it now! Nine values all told, and then I computed limits to show that there were no other values it could possibly be before (-12, 3) and after (20, 5). Thanks guys!

 

[abia ubong]

i feel u should equate xy/[x+y]as 4n/nwhich is still the samenow in this caseu might take xy 2 be 4n and x+y as n .then solve the simul eqn.n being any number,just give it a try