Proposition: All Solutions Given a Solution of an LDE With Two Variables

Is the linear Diophantine equation (LDE) $ax+by=c$ solvable according to the existence of solutions of an LDE with more variables, and is the pair of numbers $x_0,y_0$ solving this LDE, then all solutions $x,y$ given by $$x=x_0+h\frac b{\gcd(a,b)},\quad y=y_0-h\frac a{\gcd(a,b)},\quad \forall h\in\mathbb Z.$$

Proofs: 1

Proofs: 1

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
  2. Jones G., Jones M.: "Elementary Number Theory (Undergraduate Series)", Springer, 1998