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.
The equation ax + (a+2)y = c is a linear equation that can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
To solve ax + (a+2)y = c, you can use the method of substitution or elimination. In the method of substitution, you solve for one variable in terms of the other and then substitute the expression into the other equation. In the method of elimination, you manipulate the equations to eliminate one of the variables and then solve for the remaining variable.
When there exist solutions to ax + (a+2)y = c, it means that there are values for x and y that satisfy the equation. These values can be found by solving the equation using the methods mentioned in the previous question.
Yes, there can be more than one solution to ax + (a+2)y = c. In fact, there can be an infinite number of solutions if the equation represents a line. This means that any point on the line can be a solution to the equation.
Yes, it is possible for there to be no solutions to ax + (a+2)y = c. This happens when the equation represents parallel lines that never intersect. In this case, there is no point that satisfies both equations and therefore, no solution.