# Intuitive explanation of solutions to a linear diophantine equation

"Given a linear diophantine equation $$ax+by=c$$ with a particular solution $$(x_0,y_0)$$ the general solution is given by $$\biggl(x_0-\frac{b}{gcd(a,b)}t,y_0+\frac{a}{gcd(a,b)}t\biggr)$$ for all $$t\in \mathbb{Z}$$"

I understand the proof of this theorem pretty well but would appreciate an intuitive explanation of why this general solution gives all the solutions to the equation...

## 1 Answer

Let's take, as an example, the case of money. We have banknotes whose values are $$a$$ and $$b$$. We want to exchange an amount of $$c$$.

Finding a solution $$x_0, y_0$$ then corresponds to finding some number of banknotes such that we exchange exactly $$c$$.

But, sort of as a dufus, I could insist on exchanging more bills.

So if I give you $$x_0 + \frac{b}{gdc(a,b)}t$$ number of bills, valued $$a$$, and you give me $$y_0 - \frac{a}{gdc(a,b)}t$$ number of bills valued $$b$$ back, we have exchanged exactly an additional $$\frac{ab}{gcd(a,b)}t - \frac{ba}{gcd(a,b)} t = 0$$

So we would still exchange exactly $$c$$