F(x,y) = x

^{2}- 2xy + y

^{2}- 5x + 7y

G(x,y) = x

^{2}- 3xy + 2y

^{2}+ x - y

H(x,y) = xy - 12x + 15y

If x and y are integers and p is an odd prime such that the values of F(x,y) and G(x,y) are both divisible by p, then prove that H(x,y) is also divisible by p.

Seventeen divides expression? is a specific instance of this puzzle where p=17.