Show that no non-zero integers $a, b, x, y$ satisfy: \begin{cases} ax-by=16. \\ ay+bx=1. \end{cases}
From Baltic-Way 2021.
\begin{align} &(a+bi)(x+yi)=(ax-by)+i(ay+bx)=16+i. \\ &|(a+bi)(x+yi)|=|16+i|. \\ &\sqrt{a^2+b^2}\sqrt{x^2+y^2}=\sqrt{257}. \\ &(a^2+b^2)(x^2+y^2)=257. \\ &\therefore a^2+b^2=257, x^2+y^2=1 \text{ or } a^2+b^2=1, x^2+y^2=257.(\because 257: \text{ prime.)} \\ \Rightarrow & \text{No solution.} \end{align}
Is my solution right?
Your solution is indeed valid, and as FShrike writes in the comments, quite elegant. I'm putting this confirmation in a community wiki answer to mark this question as answered. (I think this is how that works!)