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"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...

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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 $c$

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