I think he introduces c more for proving (b) than proving d=gcd(a,b).(a) and , and,
(b) If there exists , such that and , then .
Now, I want clarification on proving (b). My professor last semester always did something like this. In problems where we were working with linear combinations, he would have the equation for . No problem. But to show that , he would say: "suppose there is some such that and . then and for . Thus, ."
now i always had a problem with that. why does this work? it seems like an underhanded trick to me, and that it could work for 's that are not the gcd.
Let c be a common divisor of a and b. Thus he must divides gcd(a,b), since it's the least common factor of all common divisors of a and b.
So we proved in (a) that d, the presumed gcd, divides a and b. Thus he divides the gcd. But we proved in (b) that if c is any of a and b's common divisors, then it divides d. So there is no common factor of a and b higher than d !
Does d divide a ?case in point: lets say i am thinking about (3,2), we know this is 1. and , so here . and i can run the proof above and it works for my (= 1).
but what if i was really imagining the combination , and so i took . and i can run the generic proof above for and conclude that my (which is 2) is the gcd of 3 and 2.
Plus, as soon as you can simplify the equaion am+bn=..., do it.
Because you make assumptions. It's better having an example right nowof course if we know the values of , this would be absurd, but a lot of the proofs i am doing are with pure variables. how can i be confident that my linear combination is the optimal one for the gcd and thus can run this proof with a clear conscience.
By the way, you don't have to care about m and n. Because the gcd divides any linear combination of a and b.
In proofs involving pure variables, you barely need the identity : there exist m and n such that am+bn=gcd(a,b). It is not interesting because it's not a reciprocity... Saying that the gcd divides a and b and any linear combination of them two is more interesting
Prove it as it comes in your mind lol. No, I'm serious. There is a lot of intuition in my opinion.or am i imagining a problem that really isn't there?
Are there unclear things ? (which wouldn't be surprising... not easy to explain this way )