Prove that a+b+c is a divisor of a^2+b^2+c^2

Hi,

I was hoping somebody could put me on the right line to prove this question. It says:

Let, be positive integers such that is a rational number. Prove that is a divisor of

I'm never very good at proofs as I never know where to start. Would one times by and see what happens (I have but I can't see anything helpful)?

Thanks very much in advance for any help!

Re: Prove that a+b+c is a divisor of a^2+b^2+c^2

First, note that if one of is then the other also must be If then is a divisor of

Assume and let where and

Rearrange the equation to get As the RHS is rational, the LHS would be irrational unless Hence

Finally, we have since

Re: Prove that a+b+c is a divisor of a^2+b^2+c^2

Quote:

Originally Posted by

**Sylvia104** Rearrange the equation to get

. As the RHS is rational, the LHS would be irrational unless

. Hence

Finally, we have

since

Ok right I sort-of see what's happening. Although I'm not totally sure on some steps above.

Could you just expalin the method a little more as I am slightly unsure of what you are doing?

Thank you very much for your help so far!

Re: Prove that a+b+c is a divisor of a^2+b^2+c^2

Actually, by writing I implicitly assumed that To make absolutely sure, I should check that if then which is not rational; therefore rational

Re: Prove that a+b+c is a divisor of a^2+b^2+c^2

Oh right! Thanks so much for the help! It always seems so simple when you know how.

Thanks again.