Algebra Solutions: Construct solutions to a particular Diophantine equation from a given solution

Construct solutions to a particular Diophantine equation from a given solution

Let $a, b, N$ be integers with $a,b \neq 0$ , and let $d = \mathsf{gcd}(a,b)$ . Suppose $(x_0,y_0)$ is a solution of the equation $ax + by = N$ . Prove that for any integer $t$ , $(x_0 + \frac{b}{d} t, y_0 - \frac{a}{d} t)$ is also a solution of $ax + by = N$ .

Let $x_t = x_0 + \frac{b}{d} t$ and $y_t = y_0 - \frac{a}{d} t$ . Then we have that

$a x_t + b y_t$	=	$a(x_0 + \frac{b}{d} t) + b(y_0 - \frac{a}{d} t)$
	=	$ax_0 + \frac{ab}{d} t + by_o - \frac{ab}{d} t$
	=	$ax_0 + by_0$
	=	$N$ .

Thus $(x_t,y_t)$ is a solution of $ax + by = N$ .

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