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!

Easy Math Editor

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Comments

$\frac{1}{x} + \frac{1}{y}= \frac{1}{2013} \implies xy= 2013(x+y)$ 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 - 8 years, 1 month ago

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. - 8 years, 1 month ago

Thank You!

Anoopam Mishra - 8 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$