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Let us consider a congruence of the form: $$x≡(xy) \bmod z$$

Then the question is: How one can find $y$ in terms of $x$ and $z$

player3236
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Safwane
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  • This is equivalent to $z\mid x(y-1)$. Thus, $\frac zd\mid y-1$, where $d=\gcd(x,z)$. That is all that we know about $y$, so we can't find $y$. – Rushabh Mehta Jan 26 '21 at 14:51

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Well, if $x$ is not a zero divisor in $\Bbb Z_z$, i.e., $gcd(x,z)=1$ and so $x$ is a unit in $\Bbb Z_z$, the shortening rule can be applied in $\Bbb Z_z$:

$x\cdot 1 = x\cdot y \Rightarrow y=1$

or you can multiply $x=xy$ in $\Bbb Z_z$ with $x^{-1}$.

Wuestenfux
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