Positive solution for linear Diophantine equations

[pyfgcr]

The linear Diophantine equations: ax+by=c, a,b,c is natural numbers.
If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y.
However, I wonder how to find positive integer solution x,y only.

 

[DonAntonio]

The linear Diophantine equations: ax+by=c, a,b,c is natural numbers.
If c is a multiple of gcd(a,b), there is infinite integer solutions, and I know how to find x,y.
However, I wonder how to find positive integer solution x,y only.

They may not exists. For example, the equation $\,7x+6y=5\,$ cannot have positive solutions, but it has

solutions, like $\,(5,-6)\,$

DonAntonio

 

[Mensanator]

Shouldn't that be (5,-5)?

 

[DonAntonio]

Shouldn't that be (5,-5)?

Yes, you're right of course. Thanks.

DonAntonio

 

[HallsofIvy]

All solutions of the Diophantine equation ax+ by= c (assuming a, b, relatively prime) are of the form x= x₀+ kb, y= y₀- ka for k any integer. If you want both x and y positive, you must be able to choose k so that those are positive. Whether that is possible, of course, depends on a, b, x₀, and y₀.