Originally Posted by

**Hartlw** The proof referenced in post 3 is reproduced here for convenient reference.

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“Theorem: Let a and b be relativeley prime positive integers. If c > ab, then there exist positive integers x and y such that ax + by = c.

There are integers x_{0},y_{0}, such that ax_{0} + by_{0} = c. Consequently, all integer solutions of the equation ax + by = c have the shape x = x_{0} + bt, y = y_{0} - at, where t ranges over the integers. To produce a positive solution, we want to find t such that x > 0 and y > 0.

So we need y_{0} – at > 0, x_{0} + at > 0, or equivalently

-x_{0}/b < t < y_{0}/a.*

The interval (-x_{0}/b,y_{0}/a) has width y_{0}/a + x_{0}/b which simplifies to (ax_{0} + by_{0})/ab, that is c/ab. If c/ab >1, then the interval is guaranteed to contain the integer t, and we are finished.”