Alice wants to color the integers between 2 and 31 (inclusive) using $k$ colors. She wishes that if $m$ and $n$ are integers between 2 and 31 such that $m$ is a multiple of $n$ and $m \neq n$, then $m$ and $n$ are of different colors. Determine the smallest integer $k$ for which Alice can color the integers $2,3, \ldots, 31$ using $k$ colors.
We show that $k \geqslant 4$. Consider the numbers 2, 4, 8, and 16. 2 divides the other 3, so 2 is not the same color as any of the others. Similarly, 4 divides 8 and 16, so it does not share a color with any of the others. And 8 divides 16, so they each have a different color. We have thus $k \geqslant 4$. Construction: We color the numbers from 2 to 3 in red, from 4 to 7 in blue, from 8 to 15 in green, and from 16 to 31 in yellow. We can verify that the construction works (This is because for $m$ divides $n$ and $m \neq n$, we have that $n \geqslant 2 m$).