Mixing
Elementary question about a fake-proof and greatest common divisors
Clash Royale CLAN TAG#URR8PPP
up vote
1
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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!
elementary-number-theory fake-proofs
asked 1 hour ago
Sergei Richter
265
add a comment |Â
up vote
1
down vote
favorite
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!
elementary-number-theory fake-proofs
asked 1 hour ago
Sergei Richter
265
2
You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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!
elementary-number-theory fake-proofs
asked 1 hour ago
Sergei Richter
265
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!
elementary-number-theory fake-proofs
elementary-number-theory fake-proofs
asked 1 hour ago
Sergei Richter
265
asked 1 hour ago
Sergei Richter
265
edited 1 hour ago
asked 1 hour ago
Sergei Richter
265
asked 1 hour ago
Sergei Richter
265
asked 1 hour ago
Sergei Richter
265
265
2
You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago
add a comment |Â
2
You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago
2
2
You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago
You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
2
down vote
accepted
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.
answered 1 hour ago
Eric Wofsey
172k12198318
add a comment |Â
up vote
2
down vote
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.
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
add a comment |Â
up vote
1
down vote
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!)
answered 1 hour ago
P Vanchinathan
14.3k12036
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
add a comment |Â
up vote
0
down vote
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.
answered 52 mins ago
Adam
45213
add a comment |Â
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
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.
answered 1 hour ago
Eric Wofsey
172k12198318
add a comment |Â
up vote
2
down vote
accepted
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.
answered 1 hour ago
Eric Wofsey
172k12198318
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
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.
answered 1 hour ago
Eric Wofsey
172k12198318
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.
answered 1 hour ago
Eric Wofsey
172k12198318
edited 1 hour ago
answered 1 hour ago
Eric Wofsey
172k12198318
answered 1 hour ago
Eric Wofsey
172k12198318
answered 1 hour ago
Eric Wofsey
172k12198318
172k12198318
add a comment |Â
add a comment |Â
up vote
2
down vote
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.
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
add a comment |Â
up vote
2
down vote
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.
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
add a comment |Â
up vote
2
down vote
up vote
2
down vote
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.
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
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.
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
answered 1 hour ago
Mohammad Riazi-Kermani
37k41957
37k41957
add a comment |Â
add a comment |Â
up vote
1
down vote
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!)
answered 1 hour ago
P Vanchinathan
14.3k12036
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
add a comment |Â
up vote
1
down vote
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!)
answered 1 hour ago
P Vanchinathan
14.3k12036
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
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!)
answered 1 hour ago
P Vanchinathan
14.3k12036
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!)
answered 1 hour ago
P Vanchinathan
14.3k12036
answered 1 hour ago
P Vanchinathan
14.3k12036
answered 1 hour ago
P Vanchinathan
14.3k12036
answered 1 hour ago
P Vanchinathan
14.3k12036
14.3k12036
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
add a comment |Â
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
Yes, I also did assume that I was wrong. This is why I provided only the outline sketch since I thought my thinking error would have been with my 'proof-strategy'. I translated now my complete proof and my tutors response as an edit to the original question.
– Sergei Richter
49 mins ago
add a comment |Â
up vote
0
down vote
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.
answered 52 mins ago
Adam
45213
add a comment |Â
up vote
0
down vote
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.
answered 52 mins ago
Adam
45213
add a comment |Â
up vote
0
down vote
up vote
0
down vote
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.
answered 52 mins ago
Adam
45213
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.
answered 52 mins ago
Adam
45213
edited 45 mins ago
answered 52 mins ago
Adam
45213
answered 52 mins ago
Adam
45213
answered 52 mins ago
Adam
45213
45213
add a comment |Â
add a comment |Â
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You need to be more specific with your proof. From the outline there should be no objection, so either your tutor is wrong or you made an error. We can't determine which.
– Matt Samuel
1 hour ago