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?

  • $\begingroup$ You have yet to use the fact that $x,y,z\in\{1,2,3,4,5,6,7,8,9\}$. $\endgroup$ – Servaes May 2 at 12:34
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    $\begingroup$ You are missing the fact that $X$, $Y$, and $Z$ have to be integers! Clue: the solution is $X=4$, $Y=3$, and $Z=5$. $\endgroup$ – Riccardo Sven Risuleo May 2 at 12:34
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    $\begingroup$ @RiccardoSvenRisuleo How is that a clue? $\endgroup$ – Servaes May 2 at 12:35
  • $\begingroup$ It's a clue in the sense that there is a solution; now it's just a matter of finding it and proving that it is the only one. $\endgroup$ – Riccardo Sven Risuleo May 2 at 12:36
  • $\begingroup$ Well finding it is now no longer an issue, I guess. $\endgroup$ – Servaes May 2 at 12:36

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 $y$...


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