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

#NumberTheory #Pre-RMO

Easy Math Editor

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`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}`	$\boxed{123}$

Comments

the answer is 3. take g as gcd and then keep analysing number theoritically what could be the values of g.

Abhishek Bakshi - 6 years, 8 months ago

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 - 6 years, 6 months ago

Is answer 3??

Ar Agarwal - 6 years, 8 months ago

Even I got 3 pairs. $(2,3), (2,4)$ and $(3, 3)$ . How did you solve it?

Pranshu Gaba - 6 years, 8 months ago

I think answer is 2

MAYYANK GARG - 6 years, 8 months 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}`	$\boxed{123}$