inorganic chemistry - The Crisscross method for finding the chemical formula

I am reading this wikipedia article that I don't understand. What I don't understand is:

suppose we have two elements $X$ and $Y$ having oxidation numbers $x$ and $y$ respectively. Can we prove that the compound formed will be $X_yY_x$.

I tried to understand this method as: I think this method comes from the fact of charge neutrality. Suppose the compound to be formed has formula $X_{y'}Y_{x'}$. The charge on $y'$ number of $X$ elements is $xy'$. The charge on $x'$ number of $Y$ elements is $x'y$. Now by charge neutrality we should have :

$$|x|y'=x'|y|$$
this implies $\dfrac{|x|}{|y|}=\dfrac{x'}{y'}$.

But I could not further understand why the we divide $x$ and $y$ with their greatest common factor to find the required solution.

I have also found some contradiction examples to this Crisscross rule. Example is : Solid-state-semiconductor $GaAs$, in this Gallium has valancy $3$ and $As$ has valancy $5$.

The main points on which I want answers are:

• What is the logic behind this method, how to prove this method?


• Who, how and when developed this method?


• Is this method only applicable to ionic compounds?


• Why compounds like $GaAs$ do not follow this rule, how charge neutrality is preserved in these compounds. I think these don't follow this rule because these are not ionic compounds, if yes then how the chemical formula of compounds like solid state $GaAs$ is known(or decided) and how it is justified, how one atom of $Ga$ balances one atom of $As$ ,their valencies are different, how they are bonded(covalently I guess) together?