Finding the GCD of two Gaussian integers How do you calculate the GCD of $6-17i$ and $18+i$ in $\Bbb Z [i]$?

You can use the Euclidean algorithm in $\Bbb Z[i]$: divide one number by the other to obtain a quotient and remainder, then repeat with the previous divisor and remainder, and so on.  The quotient must of course be an element of $\Bbb Z[i]$, and the remainder must have norm less than that of the divisor: to ensure this we'll take the quotient to be the exact quotient, with real and imaginary parts rounded to the nearest integer.  In this case we have
$$\frac{18+i}{6-17i}=\frac{91+312i}{325}$$
and so we take the first quotient to be $i$.  Thus
$$\eqalign{
  18+i&=(i)(6-17i)+(1-5i)\cr
  6-17i&=(\cdots)(1-5i)+(\cdots)\cr
  1-5i&=(\cdots)(\cdots)+(\cdots)\cr}$$
and so on.  See if you can finish this.
