For each integer i, let s; be the number of pairs (x,y) of integers such that 1 x 120 and 5x + 8y = i. LetS = $. Find the remainder when S is divided by 1000. x, y, and z are reals such that x + y + 2z = -12 2xy + 3xz + yz = 2y + 3z + 24 2xyz = 3 + 2yz + 2yz. the form where a,b,c,d are If the largest possible value of x can be expressed in positive integers, b is not divisible by the square of any prime, and gcd(a,c,d) = 1, find a+b+c+d. ab-c d )