Farey sequence and Graham's conjectures

Let $F_n$ be the Farey sequence of order $n$. For $S\subseteq F_n$ we let $$\mathcal{Q}$(S) = \left{x/y:x,y\in S, x\le y , , $\text{and}$ , , y\neq 0\right}$\mathrm{.}Weshowthatif$$\mathcal{Q}$(S)\subseteq F_n$,then$|S|\leq n+1$\mathrm{.}Moreover,weprovethatinanyofthefollowingcases:(1)$$\mathcal{Q}$(S)=F_n$;(2)$$\mathcal{Q}$(S)\subseteq F_n$and$|S|=n+1$, we must have $S=\{0,1,\frac{1}{2},\dots ,\frac{1}{n}\}$ or $S=\{0,1,\frac{1}{n},\dots ,\frac{n-1}{n}\}$ except for $n=4$, where we have an additional set $\{0,1,\frac{1}{2},\frac{1}{3},\frac{2}{3}\}$ for the second case. Our results are based on Graham’s GCD conjectures, which have been proved by Balasubramanian and Soundararajan.