How many positive integer pairs $(x, y)$ ?

The equation can be rewritten as $(x-12)(y-12) = 144$ . Its solutions are given by $(x,y)=(12+a,12+\frac{144}{a})$ , where $a$ is a positive divisor of $144$ . Since $144=2^4 3^2$ , it has $(4+1)\times(2+1)=15$ positive divisors, so the answer is $\boxed{15}$ .