Can a Basis Be Proven with Divisibility?

[penguin007]

Homework Statement

Hi everyone.
I'm studying a problem and I need to prove that I have a basis. I tryed a proof and to achieve it I need to show that :

if k divides a*b and also divides a² +2*b² Then k divides both a and b.

Homework Equations

I'm not sure what I'm asserting is true but if it was, then it would be great for me ( cause my proof would be finished).
I first thought that if k divided a sum then it divided every term of that sum but I understood it was wrong (the other way is correct).

I'm taking any idea!

 

[Tedjn]

No, what if a = b and k = a² in general?

 

[penguin007]

No, what if a = b and k = a² in general?

That's right indeed. But what if k is prime??

 

[Tedjn]

If k = 2, then no. Why? If k > 2 is prime, then yes. Why? Use the property that if p is prime and divides ab, then it divides a or b.

 

[penguin007]

I can see if k>2, thank you, but why is it wrong if k=2 ?

 

[Tedjn]

Let a be even and b be odd.