Problem 3 (20 points) Let a and b be positive integers. Let n be an integer drawn at random from the set S = {1, 2, . . . ,ab}, where each element has equal probability of being drawn. That is, for every 3 e S, Pr['n, = s] = i. For any integers a: and y, the notation (y DIV 3:) means that 3/ divided? .7; (i.e. x = y- z for some integer z). (a) (8 points) Let c be a positive integer that divides ab. Find Pr[(c DIV 1a)]. (b) (12 points) Find Pr[(a DIV n) | (b DIV 11)]. (Hint: The least common multiple of integers 3: and 3;, written lcm(a:, y), is the smallest positive integer 6 such that (a: DIV E) and (y DIV B). You can use the following facts. 0 1 51cm(ac,y) 5 any a lcm(3:, y) divides my. 0 For any integer z, (:17 DIV z) and (y DIV 2) if and only if (lcm(r1:,y) DIV z))