Find all solutions of $\frac{2}{x}+\frac{8}{y}=\frac13$

Asked Nov 16 '22 at 13:00

Active Nov 16 '22 at 13:15

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If $x,y\in\mathbb{N}$ where $\mathbb{N}=\{1,2,3,\cdots\}$ Find all possible solutions to the diophantine equation $$\frac{2}{x}+\frac{8}{y}=\frac13$$

I tried with $AM\ge GM$ and I got $$xy\ge576$$ Then I used the fact, $$\frac{2}{x}<\frac{1}{3}\:\:\textrm{and}\:\:\frac{8}{y}<\frac{1}{3}$$ and I got these two results $$x>6,y>24$$ but what to do next$?$

Any help is greatly appreciated.

edited Nov 16 '22 at 13:15

Bill Dubuque

272,048

asked Nov 16 '22 at 13:00

Vanessa

1,253

2

Hint. $$y=\frac {24x}{x-6}=\frac {24(x-6)+24\cdot6}{x-6}=24+\frac {24\cdot 6}{x-6}$$ – lone student Nov 16 '22 at 13:07
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Hint: $xy-6x-24x=0$ and $(x-6)(y-24)=144$ – Lion Heart Nov 16 '22 at 13:13
See the linked dupe (and its many linked questions) for the standard method(s) for solivng such equations (including those above). – Bill Dubuque Nov 16 '22 at 13:16

Find all solutions of $\frac{2}{x}+\frac{8}{y}=\frac13$

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