Recently Professor Peter Cameron posed a number theory problem which is related to graphs of groups. The problem is:

Problem:

Let $n$ be a positive integer. Show that there exist subsets $A_1, A_2, …,A_n$ of $\{1,2,…n\}$ with the properties

$|A_i| = \varphi(i)$ for $i = 1,2,…,n$;

if $\mathrm{lcm}(i,j) ≤ n$, then $A_i$ and $A_j$ are disjoint, where $\mathrm{lcm}$ denotes the least common multiple.

Professor Cameron gave some motivations for this question in his blog.

I checked it for a lot of numbers and the results seems to be true. Also, it seems that for some class of numbers there are patterns. For example if $n$ is prime and the statement is true, then it seems that it is true for the multiples of it.

**My question: Is it sufficient that we prove this problem for prime numbers?**

The first version of the paper published. Congratulate to Veronica Phan, a medical student who knows mathematics well. I think she is the first medical student which has common paper with professor Cameron. Paper