Equation Solution

From the given equation $13(x-1) = 17(y-1)$ , it is obvious the equation is true when $x=y=1$ . But $x, y > 1$ . SInce 13 and 17 are primes, for the equation to be true, $x-1$ and $y-1$ must be the multiples of 17 and 13 respectively. Therefore the smallest $x$ and $y$ greater than 1 are when $x-1=17 \implies x = 18$ and $y-1=13\implies y=14$ and $x+y = 18+14 = \boxed{32}$ .

Let 13(x-1) and 17(y-1) be a product a. note that a must have a factorization including 13, and 17 since they are both prime. The leas common multiple of 13 and 17 leads us to 13=y-1, and 17=(x-1), thus y=14, x=18 and y+x=32. No smaller products can be found since the product 13*17 is the least common multiple.