Number theory - gcd and linear diophantine equations

[reb659]

Homework Statement

Suppose that gcd(a, b) = 1 with a, b > 0, and let x0, y0 be any integer solution to the equation ax + by = c. Find a necessary and sufficient condition, possibly depending on a, b, c, x0, y0 that the equation have a solution with x > 0 and y > 0.

Homework Equations

The Attempt at a Solution

I'm pretty lost. Can anyone point me in the right direction?

 

[VKint]

Well, if one of x_0 or y_0 is negative, then you have to look for another solution (x_1, y_1) with x_1, y_1 > 0. By subtracting equations, you get a(x_1 - x_0) + b(y_1 - y_0) = 0. Can you see why this equation implies that x_1 - x_0 = kb and y_1 - y_0 = -ka for some integer k? Once you do, your problem is reduced to a search for a value of k such that x_0 + kb > 0 and y_0 - ka > 0. You should be able to take it from here.