Problem 9.7. The numbers $1, 2, 3, \ldots, 57$ are written on a board. What is the maximum number of these numbers that can be chosen so that no two chosen numbers differ exactly by a factor of 2.5?
Answer: 48.

Solution. Consider sequences of natural numbers that satisfy the following set of conditions: in each sequence

- each number does not exceed 57
- there are at least two numbers, and they all go in ascending order;
- each subsequent number is 2.5 times the previous one.

Let's list them all.

- 2,5
- $4,10,25$
- 6,15
- $8,20,50$
- 12,30
- 14,35
- 16,40
- 18,45

Notice that in each of these nine sequences, there should be at least one unselected number. Then the number of selected numbers is no more than $57-9=48$.

An example where the number of selected numbers is exactly 48 is not hard to construct: it is sufficient to select all numbers from 1 to 57, except for the second numbers from the nine sequences listed above.