Values without integer solutions in linear Diophantine Equation

Question

In a linear Diophantine Equation in the form of ax+by=n.

Is it possible to find all values of n that don't have integer values for x and/or y.

For example 7x+8y=6, x and/or y don't have integer solutions

Are the values of n that don't have integer solutions infinite?

Please this is not a homework

Am just curious.

Answer 1

If a and b have a common factor the both ax and by have that factor for all x and y so ax+ by has that factor. If n does not have that factor, the Diophantine equation ax+ by= n has no (integer) solutions. For example 2x+ 6y= 5 has no (integer) solutions.

Answer 2

$ax+by=n$ is a line, thus with the density of continuum.

If you take out the double integral (diophantine) solutions, which may be none, or countable (finite or infinite, depending on the bounds), then you are left, at the minimum, with $\mathbb R \backslash \mathbb Z$ (eventually, within the given bounds).