Problem

Source: Greece Archimedes Junior 2019

Tags: combinatorics



In the table are written the positive integers $1, 2,3,...,2018$. John and Mary have the ability to make together the following move: They select two of the written numbers in the table, let $a,b$ and they replace them with the numbers $5a-2b$ and $3a-4b$. John claims that after a finite number of such moves, it is possible to triple all the numbers in the table, e.g. have the numbers: $3, 6, 9,...,6054$. Mary thinks a while and replies that this is not possible. Who of them is right?