Question 104
Let $a_1 \lt a_2 \lt \dots \lt a_k$ be positive integers. For each $a_i$ Zarina has written down all the positive divisors of $a_i$ in the notebook (some numbers might be written several times). Then, Marina split all the numbers in the notebook into several groups. It turned out that the numbers in each of these groups form a set of all the positive divisors of some positive integer, which Marina has decided to also write down on the desk. Prove that the numbers on the desk are $a_1, a_2, \dots, a_k$ in some order.