INMO 1996 Problem 1

a) Given any positive integer $n$ , show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$ ;

b) If for some positive integers $x$ and $y$ , $x+j$ divides $y+j$ for all positive integers $j$ , prove that $x = y$ .