Find the greatest common factor of $1001$ and $2431.$
By the Euclidean algorithm for computing greatest common factors, we have  \[
\text{gcf}(1001,2431)=\text{gcf}(1001,2431-2\cdot 1001) = \text{gcf}(1001,429).
\]Applying the Euclidean algorithm again, we get \[
\text{gcf}(1001,429)=\text{gcf}(429,1001-2\cdot 429)= \text{gcf}(429,143).
\]Recognizing 429 as $3\times 143$, we conclude that the greatest common factor of 1001 and 2431 is $\boxed{143}.$