# Simple Linear Diophantine Equation - problem with proof.

I'm reading up on diophantine equations and one of the theorems is that

"if $x,y$ is any solution of $ax + by = c$, then it is of the form $x_0 +\dfrac{b}{d}t ,\, y_0 - \dfrac{a}{d}t$ where $d = \gcd(a,b)$"

(Where $(x_0, y_0)$ is a particular solution.)

They subtract the two equations, then divide by d to give $\dfrac{a}{d}(x-x_0) + \dfrac{b}{d}(y-y_0) = 0$

Then they claim that $\dfrac{b}{d}$ must divide $x-x_0$ but I don't understand why, could someone please explain?

Let $a'=\frac{a}{d}$ and $b'=\frac{b}{d}$. Then $a'(x-x_0)+b'(y-y_0)=0$. This can be rewritten as $$b'(y_0-y)=a'(x-x_0).\tag{1}$$ Since $d=\gcd(a,b)$, the numbers $a'$ and $b'$ are relatively prime. From (1) we can see that $b'$ divides $a'(x-x_0)$. Since $\gcd(b',a')=1$, it follows that $b'$ divides $x-x_0$.