How to find the highest common factor of two complex numbers

Asked May 10 '17 at 00:35

Active May 10 '17 at 00:35

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I have $$7+i,\quad\quad 5+3i\in\mathbb{Z}[i]$$

The answer is given as $$7+i=(5+3i)\cdot 1+(2-2i)$$

$$5+3i = (2-2i)\cdot 2i+(1-i)$$

$$2-2i=(1-i)\cdot 2$$

So $hcf(7+i,5+3i) = 1-i$

What I don't understand is how they're finding what to multiply by, like in the first line, they multiply $5+3i$ by $1$, in the second line they multiply $2-2i$ by $2i$. I tried using the fraction $\frac{7+i}{5+3i}$, and then rationalising the denominator but it got complicated and didn't look anything like what they've gotten. Could someone explain their steps in more detail?

asked May 10 '17 at 00:35

mrnovice

5,773

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Duplicate of GCD of gaussian integers. Also see some of the links posted there: Finding GCD of Complex Numbers with Euclidean Algorithm, GCD computations in $\mathbb{Z}[i]$. – dxiv May 10 '17 at 00:43
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@dxiv Thanks, those links solved my issues. Wasn't able to find it because I didn't know what gaussian integers were. – mrnovice May 10 '17 at 00:49
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In an ideal world, those MSE links would have been automatically listed under the related column on the RHS of this page. But apparently the MSE search function doesn't know how to find $\mathbb{Z}[i],$, either. – dxiv May 10 '17 at 00:58

How to find the highest common factor of two complex numbers

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