Solve equation $\frac{1}{x}+\frac{1}{y}=\frac{2}{101}$ in naturals

A start: Rewrite as $2xy-101x-101y=0$ and then as $4xy-202x-202y=0$ and then as $(2x-101)(2y-101)=101^2$. There are not many ways to factor $101^2$.

Remark: The approach in the OP is fine, a little more complicated. If you go through the path outlined above, you will find that a couple of solutions were missed.


$$\dfrac1x=\dfrac{2y-101}{101y}\iff x=\dfrac{101y}{2y-101}$$

If $d$ divides $2y-101,101y$

$d$ must divide $2(101y)-101(2y-101)=101^2$

So, $2y-101$ must divide $101^2$ to make $x$ an integer


Sneaky hint: Think about $\frac{1}{a}+\frac{1}{a(2a-1)}$

Tags:

Elementary Number Theory

Algebra Precalculus

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