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## Question

I don't remember math looking like this when I was in 8th grade.

How many pairs of positive integers (x,y) are solutions of the equation 1/x+2/y=3/19?

## 4 answers to this question

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My guess

I could find only 3 values:

x=7, y=133

x=19, y=19

x=247, y=13

I see that with large values of x, value for y starts to saturate above 12.

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yeah..i also came up with 2 pairs given in robo's answers.... is der a way to find the solution other than substituting numbers which i did?

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robo is correct.

We can change the equation to:

1/x = 3/19 -2/y

Now if we examine the value of X as y approaches infinity (ie, gets very large) then 2/y approaches 0 so we have 1/x=3/19 for large values. To clarify this more, you might want to graph the equation. To verify the solutions by robo, I use a spreadsheet, first solved for x getting:

x=19y/(3y-38) then used formulas in spreadsheet to solve for x for 0<Y<10000 where it is obvious that X slowly approaching 6.333... which is solution we see for x as y nears infinity. Then solved for y and setup spreadsheet for 0<x<10000 where we see large for large x, y approaches 12.666...which is the solution for y as x approaches infinity.

In both cases, I created a formula to flag integer solutions and in both scenarios arrived at robo's solution.

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Wow. Thanks everybody!

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