jmc

There is a pattern here: we notice that $9009=9000+9=9\times 1000+9\times 1=9\times 1001$, while similarly $14,014=14\times 1001$. Since $9$ and $14$ have no factors in common, it follows that the greatest common factor of $9,009$ and $14,014$ is $\boxed{1001}$.