Greatest common divisor problem help

[mafendee]

Homework Statement

Given gcd(a,b)=1, what is the gcd(a^2+b^2, a+b) where ^=square.

Homework Equations

The Attempt at a Solution

gcd(a^2+b^2, a+b) = gcd ( (a+b)^2 -2ab, a+b) which i think, we can reduce to gcd(2ab, a+b). Here is where I stucked. I am not sure how to proceed. Please could anyone help me?

 

[tiny-tim]

Welcome to PF!

Given gcd(a,b)=1 …

… we can reduce to gcd(2ab, a+b). Here is where I stucked.

Hi mafendee! Welcome to PF!

Hint: if p divides 2ab, then either …

 

[mafendee]

thank you tiny-tim,

"if p divides 2ab, then either", p|2 and/or p|a (p not divide b)
or p|2 and/or p|b. (p not divide a)
p>1 cannot divide both a,b since (a,b)=1.

so gcd(2ab, a+b)=1. is this correct?

 

[tiny-tim]

thank you tiny-tim,

"if p divides 2ab, then either", p|2 and/or p|a (p not divide b)
or p|2 and/or p|b. (p not divide a)
p>1 cannot divide both a,b since (a,b)=1.

so gcd(2ab, a+b)=1. is this correct?

hmm …

you're not being logical …

why can't it be 2?​

 

[mafendee]

you're right. I had a thinking that a and b should be even and odd which is not divisible by 2. But then it could also be the case where both of them are odds, hence a+b=even which is divisible by 2.

so, the gcd = 2.

is it correct to say that gcd(a^n+b^n, a+b) = n? (i haven't tried to prove this).

 

[tiny-tim]

you're right. I had a thinking that a and b should be even and odd which is not divisible by 2. But then it could also be the case where both of them are odds, hence a+b=even which is divisible by 2.

so, the gcd = 2.

Yes, gcd = 2 if they're both odd, and 1 if one is even …

but can you prove that properly?

is it correct to say that gcd(a^n+b^n, a+b) = n? (i haven't tried to prove this).

(try using the X² tag just above the Reply box )

Well, if n is odd, then a+b divides aⁿ + bⁿ anyway.

For n = 4, for example, you would start with gcd(2ab(2a² + 3ab + 2b²),a+b) …

what happens next?

 

[Дьявол]

And what if a=7 and b=2 ?

gcd(7,2)=1

gcd(7²+2²,7+2)=gcd(53,9)=1

Where 2 came from?

 

[tiny-tim]

And what if a=7 and b=2 ?

gcd(7,2)=1

ah … 2 is even!
…

Yes, gcd = 2 if they're both odd, and 1 if one is even …

 

[Дьявол]

Yes, gcd = 2 if they're both odd, and 1 if one is even

Ohh... I missed this one... Sorry...

Regards.

EDIT:

Here is my proof.

Let's say that p and k are integers.

If a is even and b is odd then a=2p and b=2k-1 so that a²+b²= 4p² + 4k²-4k+1 and a+b = 2p+2k-1. As we can see there isn't number which is divisible by both of the terms.

If a and b are odd then a²+b²=(2k-1)²+(2p-1)²=4k²+4p²-4k-4p+2 and a+b=2p-2k-2, so that both terms are divisible by 2 so that gcd(a²+b², a+b) = 2

 

[mafendee]

thank you to both of you. i think Дьявол has proved the cases of +a,-b and -a,-b.

"For n = 4, for example, you would start with gcd(2ab(2a2 + 3ab + 2b2),a+b) "

Here is what I have:

Let say p|2ab(2a² + 3ab + 2b²), this could be the cases where

case1: p|2 or p|a but not b, and p does not divide (2a² + 3ab + 2b²)
OR
case2: p|2 or p|b but not a, and p does not divide (2a² + 3ab + 2b²)

clearly in both cases if p|(a+b), this is the cases when p|2. so p=1,2.

If both a,b are odds. let a=2k+1, b=2p+1, then a²=4k²+4k+1, b²=4p²+4p+1, ab=4kp+2(k+p) +1.

2(4kp+2(k+p)+1)(8k²+8p²+14(k+p)+12kp+7) which is divisible by 2, so does a+b, so gcd=2.

If only one is odd, let a=2k+1, b=2p, then a²=4k²+4k+1, b²=4p², ab=2p(2k+1).

2(2p)(2k+1)(8k²+4p²+2(4k+p)+2kp+2)=8(p)(2k+1)(4k²+2p²+(4k+p)+kp+1) which is divisible by 8, but a+b=2(k+p) +1 is only divisible by 1, so gcd=1.

Am i doing this correctly?

 

[tiny-tim]

mafendee, this a = 2k or 2k+1 stuff is really longwinded.

For the original case, if p is prime and p | 2ab, then p = 2 or p| a or p | b, and p can't divide both a and b, so if it divides one of them it can't divide a+b, so gcd = gcd(2,a+b) = 1 or 2.

For the (a+b)⁴ case, I was expecting you to carry on from gcd(2ab(2a² + 3ab + 2b²),a+b), and reduce the 2a² + 3ab + 2b² even further, until you get something like gcd(2ab,a+b) again.

 

[mafendee]

tiny-tim, you are right.

from gcd(2ab(2a² + 3ab + 2b²),a+b), we have
= gcd(2ab(2a+2b)(a+b) -4ab + 3ab),a+b) = gcd(2ab(ab), a+b))

if p|2a²b² then p|2 or p|a² (p|a) or p|b² (p|b) but p does not divide both a and b at the same time so p does not divide a+b. So we have gcd(2, a+b) = 1 or 2.

For odd n, say n=3 we have gcd(a³+b³, a+b) =
gcd ((a+b)(a²+b²-ab), a+b) = a+b. So if n is odd, the gcd = a+b.

What happen if n is even, will the gcd always be 1 or 2? How to prove this, by induction or?

 

[tiny-tim]

… So if n is odd, the gcd = a+b.

Yes.

What happen if n is even, will the gcd always be 1 or 2? How to prove this, by induction or?

Try expanding (a + b)²ⁿ using the binomial coefficents …

so 1 4 6 4 1 for n = 2

1 6 15 20 15 6 1 for n = 3 etc

 

[mafendee]

tiny-tim,

for even power, (a + b)²n,

let n=1,
1 2 1
we have gcd(2ab, a+b)
here gcd=1/2

let n=2,
1 4 6 4 1
we have gcd(2ab(2a²+3ab+2b²), a+b)
here gcd=1/2

let n=3,
1 6 15 20 15 6 1
we have gcd(ab(6a⁴+15a³b+20a²b²+15ab³+6b⁴), a+b)
here gcd=1

let n=4,
1 8 28 56 70 56 28 8 1
after taking 1 out after reduction, we can factorise 2 out. So i assume the gcd here is 1/2.

But then how to proof that when is the gcd=1 and when the gcd=1/2?

 

[tiny-tim]

(just got up :zzz:)

let n=3,
1 6 15 20 15 6 1
we have gcd(ab(6a⁴+15a³b+20a²b²+15ab³+6b⁴), a+b)
here gcd=1

No, 6a⁴+15a³b+20a²b²+15ab³+6b⁴

= 6a⁴+(6+9)a³b+(9+2+9)a²b²+(6+9)ab³+6b⁴

= 2a²b² + multiple of (a+b) …

and you should be able to prove that it will always be 2a^n-1b^n-1

(i expect there's a better way of proving it, but this is the most immediately obvious one )

EDIT: ah, I see the pattern now …

it's 6 - 15 + 20 - 15 + 6

(and n = 2 was -4 + 6 - 4) …

can you see what the general pattern is, and why it must always add to 2?

(hint: consider (a - b)²ⁿ)​

 

[mafendee]

here we go again,

for even power, (a + b)²ⁿ,

let n=1, we have (a + b)²
1 2 1
we have gcd(2ab, a+b)
here gcd=1/2

let n=2,
1 4 6 4 1
we have gcd(2ab(2a²+3ab+2b²), a+b) = gcd (2ab((2a+2b)(a+b) -ab), a+b) = gcd (2a²b², a+b)
here gcd=1/2

let n=3,
1 6 15 20 15 6 1
we have
=gcd(ab(6a⁴+15a³b+20a²b²+15ab³+6b⁴), a+b)
=gcd(ab((6a²+6b²)(a²+b²)-12a²b²+15a³b+20a²b²+15ab³), a+b)
=gcd(ab(((6a+6b)(a+b)-12ab)((a+b)(a+b)-2ab)+15a³b+8a²b²+15ab³), a+b)
=gcd(ab(24a²b²+15a³b+8a²b²+15ab³)), a+b)
= gcd(a²b²(15a²+32ab+15b²), a+b)
= gcd(a²b²((15a+15b)(a+b)-30ab+32ab), a+b)
= gcd(a²b²(2ab), a+b)
= gcd(2a³b³, a+b)

here gcd=1/2

but what I've got here is the gcd is of the form (2aⁿbⁿ, a+b) different from what you pointed out (2a^n-1b^n-1, a+b). am i missing something here?

The '2' that we managed to factor out after lengthy algebra there goes parallel to what you wrote on your 'edit' part. When we "sum" up the term, it will end up with 2. (6 - 15 + 20 - 15 + 6 = 2, -4 + 6 - 4 = 2 and so on).

 

[mafendee]

tiny-tim,

was my answer in previous post correct?

 

[tiny-tim]

tiny-tim,

was my answer in previous post correct?

Hi mafendee!
Sorry … I somehow missed replying to this thread.

Yes, it's correct, but a little long-winded …

you get more marks in the exam if you can make the proofs shorter, and so that they clearly work for any n.

but what I've got here is the gcd is of the form (2aⁿbⁿ, a+b) different from what you pointed out (2a^n-1b^n-1, a+b). am i missing something here?

No, I just forgot what "n" was.

The '2' that we managed to factor out after lengthy algebra there goes parallel to what you wrote on your 'edit' part. When we "sum" up the term, it will end up with 2. (6 - 15 + 20 - 15 + 6 = 2, -4 + 6 - 4 = 2 and so on).

Yup … and can you do a simple proof why that works for any n?

 

[mafendee]

tiny-tim,

i will try to work on that simple proof you said. thank you.

 

[tiny-tim]

Hi mafendee!

In case you haven't got it yet, here's a hint:

expand (1 - 1)ⁿ ​