Question 61
Kati writes the numbers
$$
2^0, 2^1, 2^2, \dots, 2^{100}, 3^0, 3^1, 3^2, \dots, 3^{100}, 6^0, 6^1, 6^2, \dots, 6^{100}
$$
on the board. In each step, she performs one of the following operations:
(1) She can pick two numbers and replace them with their greatest common divisor and least common multiple; or
(2) She can pick two numbers, one of which is divisible by the other, and replace them with some two numbers whose greatest common divisor and least common multiple would be the two picked numbers.
Find the least and the greatest sum of all numbers on the board that can be achieved via finitely many steps.
$$
2^0, 2^1, 2^2, \dots, 2^{100}, 3^0, 3^1, 3^2, \dots, 3^{100}, 6^0, 6^1, 6^2, \dots, 6^{100}
$$
on the board. In each step, she performs one of the following operations:
(1) She can pick two numbers and replace them with their greatest common divisor and least common multiple; or
(2) She can pick two numbers, one of which is divisible by the other, and replace them with some two numbers whose greatest common divisor and least common multiple would be the two picked numbers.
Find the least and the greatest sum of all numbers on the board that can be achieved via finitely many steps.