Say we have the division algorithm
Where X,Y,Z represent a non-zero digit and the remainder is Y. What is the three-digit number XYZ?
From what I gather, I re-arranged the division into an equation: $$100x+10y+z = 8x+y+80z$$
Which simplifies into $$92x+9y = 79z$$ This equation is unhelpful, since it contains three variables. What other equations can I derive to find the value of each pro numeral? Should I consider long division properties to find more expressions?
You haven't yet used the fact that $x,y,z\in\{1,2,3,4,5,6,7,8,9\}$, which greatly restricts the possibilities.
For example, reducing the equation you found modulo $79$ shows that $$13x+9y\equiv0\pmod{79},$$ where $13\times(-6)\equiv1\pmod{79}$, so this shows that $54y\equiv x\pmod{79}$. Then there are very few options left for $x$ and $y$...