I am asking myself:
How can I show that a linear equation $xm + yn = z$, for $x,y,z \in \mathbb{Z}$, cannot have a unique solution $(m,n) \in \mathbb{Z} \times \mathbb{Z}$?
2026-03-28 02:21:02.1774664462
no unique solution - linear equation
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There are many solutions if these is one solution. let $x_0, y_0$ be the unique solution of $mx_0+ny_0=z$. Then there are infintely many solutuins given as $$x=x_0+nj,~ y=y_0-jm,~ j \in Z.$$ Or no solution if $x_0, y_0$ do not exist.