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Pre-RMO 2014/11

For natural numbers \(x\) and \(y\), let \((x, y)\) denote the greatest common divisor of \(x\) and \(y\). How many pairs of natural numbers \(x\) and \(y\) exist with \(x \leq y\) satisfy the equation \(xy = x + y + (x,y)\)?

This note is part of the set Pre-RMO 2014

Note by Pranshu Gaba
2 years ago

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The answer is 3 i.e. (2,3) (2,4) (3,3) I did this by hit and trial method as I had realised that there won't be any pair which will have any number greater than 4 in it. So it took around 2 minutes to solve it Mihir Chakravarti · 1 year, 10 months ago

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the answer is 3. take g as gcd and then keep analysing number theoritically what could be the values of g. Abhishek Bakshi · 2 years ago

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I think answer is 2 Mayyank Garg · 2 years ago

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Is answer 3?? Ar Agarwal · 2 years ago

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@Ar Agarwal Even I got 3 pairs. \((2,3), (2,4) \) and \((3, 3)\). How did you solve it? Pranshu Gaba · 2 years ago

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