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The problem was to show that the gcd(a,b) = gcd(a,b+a).

We know from the textbook (I'm allowed to do this; I cited the theorem) that if a|b and a|c then a|b+c. From that we then know that gcd(a,b) is A divisor of (a,b+a), but we don't yet know it is the greatest divisor.

*I believe my mistake is somewhere in the next line of reasoning:*However we also know that gcd(a,b+a), whatever it is, cannot exceed gcd(a,b), because if it did then it would be a divisor of a (and by symmetry in the argument, a divisor of b) greater than the gcd(a,b), which contradicts the definition. It also cannot be less than gcd(a,b) because then it is no longer the greatest common divisor of (a,b+a). Thus gcd(a,b) = gcd(a,b+a).

Thank you very much, and I'll be happy to clarify if I need to.