If $x$ is a positive rational number, show that $x$ can be uniquely expressed in the form \[x=a_{1}+\frac{a_{2}}{2!}+\frac{a_{3}}{3!}+\cdots,\] where $a_{1}a_{2},\cdots$ are integers, $0 \le a_{n}\le n-1$ for $n>1$, and the series terminates. Show also that $x$ can be expressed as the sum of reciprocals of different integers, each of which is greater than $10^{6}$.