Hi and thanks for your answer.
Could you give me an example of where the Euclid's algorithm can be used,
When using the Euclidean algorithm you repeating apply the division algorithm in several steps until you get rid of the remainder. The problem here is that first application already produces no remainder. That is why this is not a good example.
Should they be polynomials of higher degree?
Look at this.
It has examples for both numbers and polynomials.
Both are extremely cases similar.