$\displaystyle r_{m-2} = q_{m} r_{m-1} + r_{m} \hspace{50} 0 < r_{m} < r_{m-1} \\* r_{m-1} = q_{m+1} r_{m} + 0$

(ignore the indentation on the first line)

"The final equation shows that $\displaystyle r_{m}$ is a factor of $\displaystyle r_{m-1}$, and then the penultimate equation shows that $\displaystyle r_{m}$ is also a factor of $\displaystyle r_{m-2}$. Continuing in this way, we find that $\displaystyle r_{m}$ is a factor of all the remainders,"

But isn't this obviously not true?

$\displaystyle r_{m-3} = q_{m-1}r_{m-2} + r_{m-1}$

So in the very next line $\displaystyle r_{m}$ is no longer a factor.

What am I not seeing?