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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, 5 months 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 · 2 years, 3 months ago

the answer is 3. take g as gcd and then keep analysing number theoritically what could be the values of g. · 2 years, 5 months ago

I think answer is 2 · 2 years, 5 months ago

Is answer 3?? · 2 years, 5 months ago

Even I got 3 pairs. $$(2,3), (2,4)$$ and $$(3, 3)$$. How did you solve it? · 2 years, 5 months ago