## Task A-1.6.

Borna wants to color each of the numbers $2,3, \ldots, 32$ with one of $k$ colors $(k \in \mathbb{N})$ such that no number is a multiple of another number of the same color. Determine the smallest natural number $k$ for which Borna can achieve this.
## Solution.

Notice that the numbers $2, 4, 8, 16,$ and 32$ must be colored with different colors according to the condition of the problem.

Therefore, at least 5 colors are needed.

It remains to prove that $k=5$ colors are sufficient. One possible coloring is:

- Color 1: numbers 2, 3
- Color 2: numbers $4, 5, 6, 7$
- Color 3: numbers $8, 9, \ldots, 15$
- Color 4: numbers $16, 17, \ldots, 31$
- Color 5: number 32.

Note: Various other colorings of the numbers in five colors are possible, and it is necessary to either explicitly state the color for each number as in the example or describe the coloring in words. For example, in color $k$, we can color all numbers that have exactly $k$ (not necessarily distinct) prime factors (the number 12 would be colored in color 3).