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Algebra Question

Find all the solutions of 1/x + 1/y = 1/2013.

I have found 8 solutions but one of my friend claims that it has 14 solutions. He told me that there is a trick for this.

Do any of you know the trick or help me find 14 solutions?

Thanks everyone!

Note by Anoopam Mishra
3 years, 10 months ago

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\( \frac{1}{x} + \frac{1}{y}= \frac{1}{2013} \implies xy= 2013(x+y) \) This implies \( (x-2013)(y-2013) = xy - 2013(x+y) + 2013^2 = 2013^2 \).
Now note that \(2013^2= 3^2*11^2*61^2\). So number of divisors (positive and negative both included) of \(2013^2\) is \(2(2+1)(2+1)(2+1) = 54 \). For each divisor of \(2013^2\) we get 1 possible value of \( ((x-2013), (y-2013)) \). However note that \( (x, y)= (0, 0) \) satisfies \( xy= 2013(x+y) \), but not the original equation. So this case also has to be excluded. Thus from the total number of solutions we should subtract \(1\). That would give the total number of solutions= \( 54-1= 53 \). Sreejato Bhattacharya · 3 years, 10 months ago

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@Sreejato Bhattacharya There are actually 53 distinct integer ordered pairs \( (x,y) \) that satisfy the equation, not 52. You might want to double check your reasoning. Hero P. · 3 years, 10 months ago

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@Sreejato Bhattacharya Thank You! Anoopam Mishra · 3 years, 10 months ago

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