Why does the Diophantine equation $6xy-3x-2y+1=0$ have no solution in $ \mathbb{Z}^2$?

Question

Consider the function $f(x,y)= 6xy-3x-2y+1$.

Question: Why does the Diophantine equation $f(x,y)=0 $ have no solution in $ \mathbb{Z}^2$?

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I have seen a few questions like this on this page..but still couldn't figure out what the trick is here..

appreciate any help!

Answer 1

$f(x,y)=(3x-1)(2y-1)$ is this is $0$ only when $x=\frac 1 3$ or $y=\frac 1 2$.