# Find integer solutions to 1/x + 1/y = 1/14

• Jun 16th 2011, 09:03 AM
VinceW
Find integer solutions to 1/x + 1/y = 1/14
Find all solutions where x and y are integers:

$\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{14}$

this can be rearranged to:

$\displaystyle \frac{xy}{x + y} = 14$

I know how to solve diophantine equations of the form

$\displaystyle ax + by = c$

Obviously the given equation is in a different form. How do I solve?
• Jun 16th 2011, 10:12 AM
abhishekkgp
Re: Find integer solutions to 1/x + 1/y = 1/14
Quote:

Originally Posted by VinceW
Find all solutions where x and y are integers:

$\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{14}$

this can be rearranged to:

$\displaystyle \frac{xy}{x + y} = 14$

I know how to solve diophantine equations of the form

$\displaystyle ax + by = c$

Obviously the given equation is in a different form. How do I solve?

$\displaystyle \frac{1}{14}= \frac{1}{x}+ \frac{1}{y}< \frac{1}{|x|}+ \frac{1}{|y|}$. Let $\displaystyle |x|<|y|$ so $\displaystyle \frac{1}{14}<\frac{1}{|x|} + \frac{1}{|y|} < 2 \frac{1}{|x|}$ which gives $\displaystyle |x|< 28$. now there are only 56 values of $\displaystyle x$ to be fed into the equation and see which ones give an integral value of $\displaystyle y$. the computation can be further reduced by using some divisibility by $\displaystyle 7$ and $\displaystyle 2$ which are the factors of $\displaystyle 14$.
• Jun 16th 2011, 10:26 AM
abhishekkgp
Re: Find integer solutions to 1/x + 1/y = 1/14
i just found another way. put $\displaystyle x=14+a, \, y=14+b$, we immediately get $\displaystyle ab=14^2$.
• Jun 16th 2011, 12:16 PM
Also sprach Zarathustra
Re: Find integer solutions to 1/x + 1/y = 1/14
Quote:

Originally Posted by VinceW
Find all solutions where x and y are integers:

$\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{14}$

this can be rearranged to:

$\displaystyle \frac{xy}{x + y} = 14$

I know how to solve diophantine equations of the form

$\displaystyle ax + by = c$

Obviously the given equation is in a different form. How do I solve?

$\displaystyle \frac{1}{x} + \frac{1}{y} = \frac{1}{14}$

$\displaystyle \frac{x+y}{xy} = \frac{1}{14}$

$\displaystyle xy = 14(x+y)$

$\displaystyle xy-14x-14y+196=196$

$\displaystyle (x-14)(y-14) = 196$