If I have the expression $\gcd(a+b,a-c)$ is there a way to further reduce this? Are there any other properties?
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No, there is no way to "simplify" the expression, since without further specification, we must take $a, b, c$ to be arbitrary integers.
Theorem: $\gcd(a,b)=\gcd(a,b-ma)$ for any integer $m$.
Try and prove that, and then use it to answer your question.
Note, though that there are no simpler ways to express this using fewer letters. But you can express it in different ways. For example, $\gcd(a+b,a-c)=\gcd(a+b,b+c)$ (Why?)