Given a lattice point grid, for a line y = -x + p, where p is a prime, why is it true that each point on the line is visible from the origin? Another way of representing this problem is to ask why the gcd(p - n, n) = 1 for some prime p and integer n? Through my research, I know that the totient function will say how many points there will be on the line, but I don't see how to show that these coordinates have to be relatively prime, and thus visible from the origin.
Edit: I meant to include that the x and y intercepts on the line are not to be included in this. Also, I did not realize that two numbers that sum to a prime must be coprime, I believe that is what I needed, thanks!