How do you find the range, algebraically, of a multivariable function?

I'm attempting to prove the following:

Prove that there do not exist integers m and n such that 12m+15n=1.

I know there's a different way to do it, but I want to show that 1 is not in the range of the multivariable function f(m,n)=12m+15n.

I guess I could simply say, "Obviously, 1 is not in the range of this function" but I want to show it algebraically for the proof.

Any help would be appreciated!