The discussion revolves around the conditions under which solutions exist for the linear Diophantine equation ax + (a+2)y = c, focusing on the role of the greatest common divisor (GCD) in determining solvability.
The conversation includes various attempts to clarify the conditions for solutions and the nature of the GCD. Some participants suggest that additional context about the variables is needed, while others propose exploring the implications of the GCD further. There is no explicit consensus, but several productive lines of inquiry are being pursued.
Participants note that a and b are natural numbers, while x and y are integers, with constraints on the GCD. There is uncertainty regarding the specific forms of the variables and their relationships.
mfb said:gcd(a, (a+2)) has a shorter expression.
Not sure what the last three lines are saying.
Shackleford said:Homework Statement
When will there exist solutions to ax + (a+2)y = c?
Homework Equations
GCD
The Attempt at a Solution
Of course, this is a linear Diophantine equation. I know that a solution will exist when gcd(a, (a+2)) | c.
gcd(a, (a+2)) = d ⇒ ∃ x,y ∈ ℤ ∋ xa + y(a+2) = d
a = 0(a+2) + a
a+2 = (a) + 2
a = β(2) + r
Ray Vickson said:Have you forgotten to tell us whether or not a, c, x, y are positive (non-negative?) integers?
I think you should, as the final expression is very easy. It is a bit like leaving 4+5 in a final answer. It is not wrong in terms of mathematics, but you should replace it by 9.Shackleford said:I just started the algorithm for finding the GCD, but I don't think that I actually need to find an expression.