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

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TopNewestThe 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 · 2 years, 9 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, 11 months ago

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I think answer is 2 – Mayyank Garg · 2 years, 11 months ago

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Is answer 3?? – Ar Agarwal · 2 years, 11 months ago

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– Pranshu Gaba · 2 years, 11 months ago

Even I got 3 pairs. \((2,3), (2,4) \) and \((3, 3)\). How did you solve it?Log in to reply