ZA-SAMO 2018 Problem 6

Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$ . Construct an $n \times 3$ table where the entries of the $k$ -th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$ . Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.

[list=a] [] Prove that at least three entries remain at the end of the procedure. [] Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain. [/list]