Let $n$ integers on the real axis be colored. Determine for which positive integers $k$ there exists a family $K$ of closed intervals with the following properties: i) The union of the intervals in $K$ contains all of the colored points; ii) Any two distinct intervals in $K$ are disjoint; iii) For each interval $I$ at $K$ we have ${{a}_{I}}=k.{{b}_{I}}$, where ${{a}_{I}}$ denotes the number of integers in $I$, and ${{b}_{I}}$ the number of colored integers in $I$.