Proof Verification- Prove that $(a+b,a-b) geq (a,b)$ for any two integers.
$begingroup$
I came across this question while solving the book Challenge and Thrill of Pre-College mathematics. Please check if the proof that I have done is correct. (I am familiar with only basic number theory including basic theory of primes and modular arithmetic.)
I believe that I might have made some mistake in reasoning while proving that $bx-aygeq 0$
Proof:
Without loss of generality, we can assume that $a$ and $b$ are both positive with $ageq b gt 0$
Since GCD can be written as linear combination of the two integers,
$$(a,b) = ax + by$$
We know that $(a,b) leq b leq a$
Which implies that either $x$ or $y$ is is negative.
We can assume that $y$ is negative. This implies $bx-aygeq 0$. (Conversely if $x$ is negative then $ay-bx geq 0$ )
Now,
$$ax + by = (a,b)$$
$$Rightarrow ax + by + bx - ay geq (a,b)$$
$$Rightarrow (a+b)x + (b-a)y geq (a,b)$$
$$Rightarrow (a+b)x + (-y)(a-b) geq (a,b)$$
$$Rightarrow (a+b, a- b) geq (a,b)$$
QED.
number-theory proof-verification greatest-common-divisor
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
$endgroup$
add a comment |
$begingroup$
I came across this question while solving the book Challenge and Thrill of Pre-College mathematics. Please check if the proof that I have done is correct. (I am familiar with only basic number theory including basic theory of primes and modular arithmetic.)
I believe that I might have made some mistake in reasoning while proving that $bx-aygeq 0$
Proof:
Without loss of generality, we can assume that $a$ and $b$ are both positive with $ageq b gt 0$
Since GCD can be written as linear combination of the two integers,
$$(a,b) = ax + by$$
We know that $(a,b) leq b leq a$
Which implies that either $x$ or $y$ is is negative.
We can assume that $y$ is negative. This implies $bx-aygeq 0$. (Conversely if $x$ is negative then $ay-bx geq 0$ )
Now,
$$ax + by = (a,b)$$
$$Rightarrow ax + by + bx - ay geq (a,b)$$
$$Rightarrow (a+b)x + (b-a)y geq (a,b)$$
$$Rightarrow (a+b)x + (-y)(a-b) geq (a,b)$$
$$Rightarrow (a+b, a- b) geq (a,b)$$
QED.
number-theory proof-verification greatest-common-divisor
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
$endgroup$
$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23
add a comment |
$begingroup$
I came across this question while solving the book Challenge and Thrill of Pre-College mathematics. Please check if the proof that I have done is correct. (I am familiar with only basic number theory including basic theory of primes and modular arithmetic.)
I believe that I might have made some mistake in reasoning while proving that $bx-aygeq 0$
Proof:
Without loss of generality, we can assume that $a$ and $b$ are both positive with $ageq b gt 0$
Since GCD can be written as linear combination of the two integers,
$$(a,b) = ax + by$$
We know that $(a,b) leq b leq a$
Which implies that either $x$ or $y$ is is negative.
We can assume that $y$ is negative. This implies $bx-aygeq 0$. (Conversely if $x$ is negative then $ay-bx geq 0$ )
Now,
$$ax + by = (a,b)$$
$$Rightarrow ax + by + bx - ay geq (a,b)$$
$$Rightarrow (a+b)x + (b-a)y geq (a,b)$$
$$Rightarrow (a+b)x + (-y)(a-b) geq (a,b)$$
$$Rightarrow (a+b, a- b) geq (a,b)$$
QED.
number-theory proof-verification greatest-common-divisor
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
$endgroup$
I came across this question while solving the book Challenge and Thrill of Pre-College mathematics. Please check if the proof that I have done is correct. (I am familiar with only basic number theory including basic theory of primes and modular arithmetic.)
I believe that I might have made some mistake in reasoning while proving that $bx-aygeq 0$
Proof:
Without loss of generality, we can assume that $a$ and $b$ are both positive with $ageq b gt 0$
Since GCD can be written as linear combination of the two integers,
$$(a,b) = ax + by$$
We know that $(a,b) leq b leq a$
Which implies that either $x$ or $y$ is is negative.
We can assume that $y$ is negative. This implies $bx-aygeq 0$. (Conversely if $x$ is negative then $ay-bx geq 0$ )
Now,
$$ax + by = (a,b)$$
$$Rightarrow ax + by + bx - ay geq (a,b)$$
$$Rightarrow (a+b)x + (b-a)y geq (a,b)$$
$$Rightarrow (a+b)x + (-y)(a-b) geq (a,b)$$
$$Rightarrow (a+b, a- b) geq (a,b)$$
QED.
number-theory proof-verification greatest-common-divisor
number-theory proof-verification greatest-common-divisor
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
asked Jan 6 at 7:13
Naman KumarNaman Kumar
12613
12613
$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23
add a comment |
$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23
$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Let $d=gcd(a+b,a-b)$ and $gcd(a,b)=e$.
$e$ must divide the sum and the difference of $a$ and $b$. This means that $e mid a+b$ and $e mid a-b$. Thus $e$ is a common divisor of $a+b$ and $a-b$. Thus $e mid gcd(a+b, a-b)=d$. Thus
$$e leq d.$$
Your question is answered.
However one can do a bit more:
Furthermore, $d mid a+b$ and $d mid a-b$. Then $d mid 2a$ and $d mid 2b$. Thus $d$ is a common divisor of $2a$ and $2b$. Consequently, $d mid gcd(2a,2b)$. But $gcd(2a,2b)=2gcd(a,b)=2e$. Thus,
$$d mid 2e implies d leq 2e.$$
So we can say
$$ e leq d leq 2e.$$
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
$endgroup$
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
add a comment |
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1 Answer
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$begingroup$
Let $d=gcd(a+b,a-b)$ and $gcd(a,b)=e$.
$e$ must divide the sum and the difference of $a$ and $b$. This means that $e mid a+b$ and $e mid a-b$. Thus $e$ is a common divisor of $a+b$ and $a-b$. Thus $e mid gcd(a+b, a-b)=d$. Thus
$$e leq d.$$
Your question is answered.
However one can do a bit more:
Furthermore, $d mid a+b$ and $d mid a-b$. Then $d mid 2a$ and $d mid 2b$. Thus $d$ is a common divisor of $2a$ and $2b$. Consequently, $d mid gcd(2a,2b)$. But $gcd(2a,2b)=2gcd(a,b)=2e$. Thus,
$$d mid 2e implies d leq 2e.$$
So we can say
$$ e leq d leq 2e.$$
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
$endgroup$
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
add a comment |
$begingroup$
Let $d=gcd(a+b,a-b)$ and $gcd(a,b)=e$.
$e$ must divide the sum and the difference of $a$ and $b$. This means that $e mid a+b$ and $e mid a-b$. Thus $e$ is a common divisor of $a+b$ and $a-b$. Thus $e mid gcd(a+b, a-b)=d$. Thus
$$e leq d.$$
Your question is answered.
However one can do a bit more:
Furthermore, $d mid a+b$ and $d mid a-b$. Then $d mid 2a$ and $d mid 2b$. Thus $d$ is a common divisor of $2a$ and $2b$. Consequently, $d mid gcd(2a,2b)$. But $gcd(2a,2b)=2gcd(a,b)=2e$. Thus,
$$d mid 2e implies d leq 2e.$$
So we can say
$$ e leq d leq 2e.$$
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
$endgroup$
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
add a comment |
$begingroup$
Let $d=gcd(a+b,a-b)$ and $gcd(a,b)=e$.
$e$ must divide the sum and the difference of $a$ and $b$. This means that $e mid a+b$ and $e mid a-b$. Thus $e$ is a common divisor of $a+b$ and $a-b$. Thus $e mid gcd(a+b, a-b)=d$. Thus
$$e leq d.$$
Your question is answered.
However one can do a bit more:
Furthermore, $d mid a+b$ and $d mid a-b$. Then $d mid 2a$ and $d mid 2b$. Thus $d$ is a common divisor of $2a$ and $2b$. Consequently, $d mid gcd(2a,2b)$. But $gcd(2a,2b)=2gcd(a,b)=2e$. Thus,
$$d mid 2e implies d leq 2e.$$
So we can say
$$ e leq d leq 2e.$$
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
$endgroup$
Let $d=gcd(a+b,a-b)$ and $gcd(a,b)=e$.
$e$ must divide the sum and the difference of $a$ and $b$. This means that $e mid a+b$ and $e mid a-b$. Thus $e$ is a common divisor of $a+b$ and $a-b$. Thus $e mid gcd(a+b, a-b)=d$. Thus
$$e leq d.$$
Your question is answered.
However one can do a bit more:
Furthermore, $d mid a+b$ and $d mid a-b$. Then $d mid 2a$ and $d mid 2b$. Thus $d$ is a common divisor of $2a$ and $2b$. Consequently, $d mid gcd(2a,2b)$. But $gcd(2a,2b)=2gcd(a,b)=2e$. Thus,
$$d mid 2e implies d leq 2e.$$
So we can say
$$ e leq d leq 2e.$$
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
edited Jan 6 at 12:27
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
answered Jan 6 at 7:33
Anurag AAnurag A
26k12250
26k12250
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
add a comment |
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
I am unsure of what you exactly proved. Did you really prove that the gcd of $a-b,a+b$ is always the double of the gcd of $a,b$? What happens when, say, $a=3, b=12$?
$endgroup$
– Mindlack
Jan 6 at 8:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
$d | gcd(2a,2b)$ doesn’t imply that $d = gcd(2a,2b)$.
$endgroup$
– Jonas De Schouwer
Jan 6 at 10:44
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
$begingroup$
Thank you for pointing out the error. I had typed some stuff which got deleted by mistake and I didn’t realize that. I have fixed it. Let me know if something is missing.
$endgroup$
– Anurag A
Jan 6 at 12:29
add a comment |
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$begingroup$
Your proof only shows that some linear combination of $a+b$ and $a-b$ is greater than or equal to $gcd(a,b)$. It doesn't necessarily imply that $gcd(a+b,a-b) geq gcd(a,b)$.
$endgroup$
– Anurag A
Jan 6 at 7:17
$begingroup$
Thank you. I suspected there was some mistake in the reasoning. If I want this question answered do I need to ask another question? Because I have no idea what to do and I am unable to find the answer anywhere.
$endgroup$
– Naman Kumar
Jan 6 at 7:23