Proposition: A Linear Term for 1 Using Two Co-prime Coefficients

Given two co-prime integers $a,b$, there exist integers $x,y$ such that the linear equation $ax+by=1$ holds. If $x_0,y_0$ is any solution, then all solutions have the form $$x=x_0+hb\wedge y=y_0-hb\quad\forall h\in\mathbb Z.$$

Notes

Proofs: 1


Thank you to the contributors under CC BY-SA 4.0!

Github:
bookofproofs


References

Bibliography

  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