Find the continued fraction for $x=\displaystyle{\frac{22241739}{19848039}}$. Now compute the $\gcd(22241739,19848039)$ using the Euclidean Algorithm. Do you notice anything interesting? Make a conjecture, and test it against other examples.

The continued fraction we seek is given by $[1;8,3,2,2,1,14,3,9]$.

Comparing this with the work done in computing the greatest common divisor, we discover that the list of quotients found exactly matches the values seen in the continued fraction.

Careful consideration of the divisions happening in both processes should reveal that this was no coincidence -- it must always happen.