**18**

Linear diophantine equations

In this chapter we present a striking application of the Euclidean algorithm. The following very ancient problem will be solved.

**Problem 18.0.1** *Given integers a, b and c, find all integers m and n such that*

Notice that Proposition 4.1.1 shows that for certain choices of *a, b* and *c* there are no solutions at all since it was proved there that there are no solutions for *a* = 14, *b* = 20, *c* = 101. The first part of the solution to the problem is to give a necessary and sufficient condition for a solution to exist. It turns out that the idea used in proving Proposition 4.1.1 gives such a condition although the proof ...

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