Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

Find all pairs

Find all pairs (a,b) of integers

such that

Last edited by tony123 (2009-09-04 12:39:25)

Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: Find all pairs

I don't believe there are any integers because...

Last edited by Fruityloop (2009-09-05 13:32:52)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Find all pairs

Hi;

There are at least 2 solutions in Z

a=18 , b=3
a=3  ,  b=18

There may be more...

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Fruityloop

Member

Registered: 2009-05-18

Posts: 143

Re: Find all pairs

Good job Bobbym!

Oops...boy was I wrong.
(a+b)(a^2-ab+b^2)=3(2007-ab) doesn't mean (a+b)=3 or (a^2-ab+b^2)=3

Last edited by Fruityloop (2009-09-06 00:54:52)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Find all pairs

Hi Fruityloop;

I am stuck on this one too.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Find all pairs

Hi Tony123;

Prove:

That shows that 6021 can be factored into an integer square and another term. Also there are no negative solutions (If b were negative b^2 would of course be positive)

We could either try all 8 factors 2 at a time or notice that one factor has to be a square. Only 1 and 9 are possible. 1 doesn't work but 9 does. b^2 = 9 -> b=3.
The only solutions are:

3,18 and 18,3.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.