Problem

Source: 45th International Tournament of Towns, Junior A-Level P3, Fall 2023

Tags: combinatorics



3. Let us call a bi-squared card $2 \times 1$ regular, if two positive integers are written on it and the number in the upper square is less than the number in the lower square. It is allowed at each move to change both numbers in the following manner: either add the same integer (possibly negative) to both numbers, or multiply each number by the same positive integer, or divide each number by the same positive integer. The card must remain regular after any changes made. What minimal number of moves is sufficient to get any regular card from any other regular card? Alexey Glebov