FIND OUT ALL POSITIVE INTEGER SOLUTIONS

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MATHPRITOM
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FIND OUT ALL POSITIVE INTEGER SOLUTIONS

Unread post by MATHPRITOM » Tue Feb 07, 2012 12:21 am

Find all the positive integer solution of the equation 3(xy+yz+zx)=4xyz.

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Nadim Ul Abrar
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Re: FIND OUT ALL POSITIVE INTEGER SOLUTIONS

Unread post by Nadim Ul Abrar » Tue Feb 07, 2012 4:20 pm

$(x,y,z)=(2,2,3)(1,4,12)(1,6,6)$ and permutations
$\frac{1}{0}$

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Masum
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Re: FIND OUT ALL POSITIVE INTEGER SOLUTIONS

Unread post by Masum » Fri Feb 10, 2012 5:57 pm

MATHPRITOM wrote:Find all the positive integer solution of the equation 3(xy+yz+zx)=4xyz.
In these kind of equations, one of my usual tactics is to convert them in divisibility relations since they are easy to deal with. Re-write as \[3x(y+z)+3yz=4xyz\]
Or \[x(4yz-3y-3z)=3yz\]
This forces $y,z$ to be such that \[4yz-3y-3z|3yz\Rightarrow 4yz-3y-3z\le3yz\Rightarrow yz\le3(y+z)\Rightarrow y(z-3)\le3z\]
Now we can find $y,z$ very easily.
One one thing is neutral in the universe, that is $0$.

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