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I am currently working on a research project and it would be exceedingly useful to know what can be known about $$gcf(a+b,c)$$ when $a,b,$ and $c$ are known. I would appreciate any theorems or anything pertaining to this matter. Thank you.

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gcf(a,b,c) must divide that number –  Ethan Dec 16 '12 at 5:14
Looks like the A,B,C conjecture - recently claimed to have been proved. –  marty cohen Dec 16 '12 at 5:29
Roughly speaking nothing. –  André Nicolas Dec 16 '12 at 5:56
Your title doesn't describe well what you are asking about. Could you please improve the title? Thanks. –  Matthew Conroy Dec 16 '12 at 6:23
Note that when you just write out the name of an operator like $\operatorname{gcf}$, it gets interpreted as a juxtaposition of variable names and formatted accordingly (e.g. italicized). To get the proper formatting, you can use \operatorname{gcf}. (I usually write this mostly for aesthetic reasons, but in the present case I really thought at first that you were multiplying by $g$ and $c$.) –  joriki Dec 16 '12 at 7:53

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