Two sets of reals with $n$ elements in common

Question

Given a positive integer $n$, find real numbers $x$ and $y$ such that the sets $X=\{x,2x,3x,\ldots\}$ and $Y=\{y,y^2,y^3,\ldots\}$ have exactly $n$ elements in common.

I haven't been able to get anywhere.

Answer 1

Consider: $$x = \frac 1 {2^n}, \quad y = \frac 1 2$$ Then: $$X = \left \{ \frac 1 {2^n \strut}, \frac 1 {2^{n-1}}, \dotsc, \frac 1 {2 \strut}, 1 \right \} \cup X'$$ where $X'$ is a set of fractions having numerators different than $1$, and $$Y = \left \{ \frac 1 {2 \strut}, \frac 1 {2^2}, \dotsc, \frac 1 {2^n \strut}, \frac 1 {2^{n+1}}, \dotsc \right \}$$ Therefore $X$ and $Y$ have exactly $n$ elements in common.