A Number Theory Challenge from the Harvard-MIT Math Tournament

Kevin writes down the integers 1, 2, . . . , 15 on a blackboard. Then, he repeatedly picks two random integers (a, b) from the blackboard, erases them, and writes down gcd(a, b) and lcm(a, b). He stops when he is no longer able to change the set of numbers written on the board. Find the maximum possible sum of the numbers on the board after this process.