This is not really a "Quickie".

The solution is a bit longer, but it's quite clever.

In base $\displaystyle A$, the fraction $\displaystyle F_1 = 0.373737\hdots$ and the fraction $\displaystyle F_2 = 0.737373\hdots$

In base $\displaystyle B$, the fraction $\displaystyle F_1 = 0.252525\hdots$ and the fraction $\displaystyle F_2 = 0525252\hdots$

Find $\displaystyle A,\,B,\,F_1,\,F_2.$