Suppose we have a large fraction like this
$\frac{2002}{429}$
which we could simplify by e.g. using the euclidean algorithm to find out the GCD. but is there kind of like a "trick" to find out the most simplified fraction for this in the fastest way possible?
Not sure if there is a faster algorithm, (maybe this is a question for the CS SE is you are looking for a computer solution), but I would first write out the factors of each number that I can most clearly see: $$ \frac{2002}{429}=\frac{2\cdot1001}{3\cdot143}=\frac{2\cdot7\cdot143}{3\cdot143}=\frac{2\cdot7}{3}=\frac{14}{3} $$ and we have it. This particular example didn't have many steps, and essentially reverted to finding the GCD, but aside from a technique like this, there is no "trick" for factoring. In fact, there is no known way to quickly factorize large numbers (if you are more interested in this topic, look into Cryptography).