Show that if ax + by = p, then gcd(a,b) = 1 or gcd(a,b) = p

Question

Suppose that a,b,x,y are integers and that p is a prime.

Show that if ax + by = p, then gcd(a,b) = 1 or gcd(a,b) = p

Answer 1

Suppose that gcd(a,b)=q. This means that q/a and q/b. But then q/(ax+by) which is the same as q/p. Since p is a prime number we must have that either q=1 or q=p and the proof is complete.