Question 164
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.
a. Prove that at least three entries remain at the end of the procedure.
b. Prove that there are infinitely many possible triples that can remain at the end of the procedure for suitable choices of $n$ and $x_1, x_2, \dots, x_n$.
a. Prove that at least three entries remain at the end of the procedure.
b. Prove that there are infinitely many possible triples that can remain at the end of the procedure for suitable choices of $n$ and $x_1, x_2, \dots, x_n$.