jmc

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 $\boxed{143}.$