This video is about equations of the form \(a X + b Y = c\) where \(a,b,c\in \mathbb{Z}\) and \(X,Y\) are variables. We aim to find all integer solutions \(x, y\).

Problems:

  1. Suppose \(d = \gcd(a,b)\) and \(d\nmid c\). Prove that \(aX + bY = c\) has no integer solutions \(X,Y\).

  2. If \(d \mid c\) prove that a solution to \(ax + by = c\) exists.

  3. Solve \(15 x + 20 y = 100\)

  4. Solve \(15x - 2y = 1\)

  5. Find all positive solutions to the above equations.