When will there exist solutions to ax + (a+2)y = c?

[Shackleford]

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

 

[mfb]

gcd(a, (a+2)) has a shorter expression.
Not sure what the last three lines are saying.

 

[Shackleford]

gcd(a, (a+2)) has a shorter expression.
Not sure what the last three lines are saying.

I just started the algorithm for finding the GCD, but I don't think that I actually need to find an expression.

Hm. Well, I know that a and a+2 are both odd or even depending on a.

 

[Ray Vickson]

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

Have you forgotten to tell us whether or not a, c, x, y are positive (non-negative?) integers?

 

[Shackleford]

Have you forgotten to tell us whether or not a, c, x, y are positive (non-negative?) integers?

Partially. a, b ∈ ℕ and x, y ∈ ℤ and 1 ≤ d ≤ min{a,b}.

 

[geoffrey159]

If you want to solve mu+nv = d when m and n are relatively prime, you should be able to find a specific solution with Bezout theorem. With that specific solution, you should be able to find a general solution (u,v) without too much difficulty.

The thing is that ax +(a+2) y = c has not that form, or at least not always. Why ? Mfb gave you a good hint. Think about it

 

[mfb]

I just started the algorithm for finding the GCD, but I don't think that I actually need to find an expression.

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.