Please help with math homework

[junsugal]

Hi, I have a little trouble formulating the criterion.

here's the question:

Suppose (a,b) = 1, a > 0, b > 0, and consider the equation ax+by= c. Suppose (Xo,Yo) is some integer solution to this equation. Can you give criterion for the equation to have positive integer solution, namely a solution (x,y) such that x > 0, y > 0

What I come up so far is that c has to be positive integer and that c has to be greater than a and b. But that wouldn't ensure that x > 0 and y >0.

Please help

 

[lookagain]

Hi, I have a little trouble formulating the criterion.

here's the question:

Suppose (a,b) = 1, a > 0, b > 0, and consider the equation ax+by= c. Suppose (Xo,Yo) is some integer solution to this equation.
Can you give [a] criterion for the equation to have [at least one] positive integer solution, namely a solution (x,y) such that x > 0, y > 0[?]

What I come up so far is that c has to be positive integer and that c has to be greater than a and b. But that wouldn't ensure that x > 0 and y >0.

Please help

What if c equals any positive common multiple of a and b?

 

[HallsofIvy]

If \(\displaystyle x= x_0\) and \(\displaystyle y= y_0\) is a solution to ax+ by= c then so is \(\displaystyle x= x_0+ kb\), \(\displaystyle y= y_0+ ka\) for any integer k:
\(\displaystyle a(x_0+ kb)+ b(y_0- ka)= ax_0+ kab+ by_0- kab= ax_0+ by_0= c\). Now look at \(\displaystyle x_0+ kb> 0\), \(\displaystyle y_0- ka> 0\).