Linear Diophantine equation with three variables and a condition
Let me start by saying that I'm new to Diophantine equations and my method certainly will not be the best possible one. I'm aware that a faster method of solving this particular case exists, but I want to know if what I did was correct and how I can finish the exercise.
What I tried:
The GCD for 39 and 55 is one, so we can set aside to be a parameter, where t is an integer. We proceed to solve this as a Diophantine equation with two variables.
I now need to use the extended Euclidean algorithm to express 1 as a linear combination of 39 and 55.
I got that
Now, the solution would be
where t and s are integers. I got this from the formula that and .
Where and are the coefficients in the Euclidean algorithm, b is and and are 39 and 55, respectively.
It's obvious that , and if I put that x and y are greater than zero, I get that
and which makes which is .
I now have the whole range of integers [0,46] which of course isn't feasible to do (as I'd have to plug them into the inequalities for s and get even more cases.
How do I proceed from here? What do I do? I still have the condition but I feel like I'm missing something. Can it be really done this way, except that it's an unimaginably hefty job of testing each case?