In calculating GCD in $\mathbb Z[i]$ does it matter if we switch $a$ and $b$?

Question

I am trying to solve this problem:

Find the generator of the ideal $(47 - 13i, 53 + 56i).$ I know that I should use Euclidean Algorithm but I am wondering if it matters if I divided $a = 47 - 13i$ by $b = 53 + 56i$ or $b = 53 + 56i$ by $a = 47 - 13i$? and why?

Thanks for your help!

Answer 1

When you use the Euclidean algorithm for ordinary integers you divide by the smaller (in absolute value) number in order to get a small remainder for the next step.

The algorithm would still work if you did it the other way since the first step would have a quotient of $0$ and a remainder the smaller number.

In the Gaussian integers you measure size using the norm.

Take it from there ...