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I have a question to an excercise - for which I have a wrong solution - and I wanted to ask you to help me understand my thinking error. The excercise was as follows:

Let $a, b, n in mathbbN$. Show that $gcd(a,b) = gcd(a,b+na)$.

My solution was (in a nutshell):

• Let $d := gcd(a,b)$.


• Then I showed that $d$ is a common divisor of both $a$ and $b + na$
  


• Then I showed, that any common divisor $c$ of $a$ and $b+na$ is smaller-or-equal than $d$
  


• After that, I concluded that by definition $d = gcd(a,b+na)$, and with the definition of $d$ I would have $gcd(a,b) = gcd(a,b+na)$, so my proof was completed.

However, my tutor did not accept the proof. He showed me the 'correct' proof, in which I woould have $d := gcd(a,b)$ and $e := gcd(a,b+na)$ and then demonstrate $d leq e leq d$.

Anyway, I do understand his proof. But I still do not understand where my thinking error is. After all, if my starting line would be, let's say, $d := 9$ and I would have been able to show the steps afterwards, then I would be able to conclude $9 = gcd(a,b+na)$, no?

Any comments would be welcome, thank you very much in advance!

EDIT
Wow, you guys answered fast! Thank you very much. I will ask my tutor on thursday again. For completeness sake, I will formulate my complete proof (my original proof is in german, so maybe there will be something lost in translation, the [Reference] are references to our script).

Let $a,b,n in mathbbN$. Let $d := gcd(a,b)$. By definition it means $d mid a$ and $d mid b$, so with [Reference] the following holds true: $d mid (b + na)$. Therefor, $d$ is a common divisor of both $a$ and $b + na$. We will show now, that $d$ is the greatest common divisor.

Let $c$ be any common divisor of $a$ and $b + na$. Therefor $c$ divides $a$, and as such $c mid na$ und therefor $c mid ((b + na) - na) = b$ (again because of [Reference]). So $c$ divides $b$, and therefor $c$ is a common divisor of both $a$ and $b$. Since $d$ was the greatest common divisor of $a$ and $b$, we have $c leq d$.

By definition of the greatest common divisor we get $d = gcd(a,b+na)$.

End of Proof. My tutor wrote, that I only showed $gcd(a,b+na) leq gcd(a,b)$ and also need to show the other direction. Then he showed me the 'correct'/'ideal' proof today. But I will ask him again on thursday!

Your proof is perfectly correct. Either you have misunderstood what your tutor said, you communicated your proof to your tutor incorrectly, or your tutor is just wrong.

The outline of the proof that you have explained is correct.

More than likely the problem was with the detaile in one of the intermediate steps.

I would check each step carefully again and there is always room for skipping a point in writing a proof.

You want to know where the error is in your proof. Given no details no one can give the answer.
All we can do is give only our opinion, so do not treat this as an answer.

The sketch given is correct, but it is merely a restatement of what is required to be proved. Possible that both you and your tutor are correct, but tutor did not understand the proof and misjudged it. But given that you expect people to be able to analyze your "proof" without providing details I am inclined to believe you are wrong. (IT IS AN OPINION!)

Ok well firstly, if your tutor has not mentioned to you something by the name of the Euclidean Algorithm, then he or she isn't being very forthcoming about their level of expertise in the subject it's as simple as that. But follow the link I provided there, and apply the method to both greatest common divisor expressions, and this might be able to help you reach a better level of understanding on the subject, the process is clearly explained in the link.

Also important to note that you shouldn't worry about studying a modern day definition of an algorithm,don't start to think this has anything to do with programming or a need to learn a programming language, this algorithm was invented over 2300 years ago, there was no necessity to update java at that point.

Secondly if your tutor's final step is to demonstrate that $d leq e leq d$, then it is in fact their solution that is completely bogus and circular, since this statement is equivalent to the original equality that we are trying to prove!

I recommend purchasing a hard cover book in Analytic Number Theory, one that provides the solutions for every exercise it contains, so that if you really are unable to find a proof you are confident with, you can look at the one the book has provided.

I currently use one that I have found to be priceless, "Problems in Analytic Number Theory" Second Edition, written by M.Ram Murty. But do ask your supervisors at your college what they recommend.