CAPS 2019 Problem 5

Determine whether there exist $100$ disks $D_2,D_3,\ldots ,D_{101}$ in the plane such that the following conditions hold for all pairs $(a,b)$ of indices satisfying $2\le a< b\le 101$ :

• If $a|b$ then $D_a$ is contained in $D_b$ .
• If $\gcd (a,b)=1$ then $D_a$ and $D_b$ are disjoint.

(A disk $D(O,r)$ is a set of points in the plane whose distance to a given point $O$ is at most a given positive real number $r$ .)