If r, q are rationals, and r+q, r*q are integers, are r, q both integers?

**bumcheekcity** · October 14th 2007, 12:55 PM

Well, the title pretty much says it all. If: $r, q e Q, r+q, r*q e Z$ , then is it necessarily true that both $r, q e Z$

I've done some experimenting, and it seems to be true. I mean, 2 and 1/2 satisfy the product, but not the sum, and 1/3, 2/3 satisfy the sum but not the product, but attempts to prove it have so far failed.

I tried saying, let p, q be represented by $a/b, c/d, where a, b, c, d E R$ . Then, $p+q = (ad+bc)/bd, p*q = ac/bd$ , and as such, hcf(ac) and hcf(ad+bc) must be some multiple of hcf(bd), but it didn't get me anywhere. Does anyone have any suggestion as to the next step to go for this?

On another note, is there a sticky anywhere with advice as to the syntax of the [ MATH ] tag?

**ThePerfectHacker** · October 14th 2007, 02:54 PM

Let $a,b\in \mathbb{Q}$ so that $a+b,ab\in \mathbb{Z}$ . Define the polynomial $f(x) = (x-a)(x-b)$ . Notice that $f(x) \in \mathbb{Z}[x]$ . Since $f(x)$ can be factored as into linear factor in $\mathbb{Q}$ by Gauss' Lemma it means it can be factored into linear factors in $\mathbb{Z}$ . Thus, $x-a,x-b$ are its only factors and so by the factor theorem it means $a,b\in \mathbb{Z}$ . Q.E.D.

I guess this shows the following generalization: if $a_1,...,a_n$ are rational and $s_i (a_1,...,a_n)\in \mathbb{Z}$ where $s_i$ are the elementary symettric functions for $1\leq i \leq n$ it means $a_1,...,a_n \in \mathbb{Z}$ .
So in particular if $a+b+c,ab+ac+bc,abc\in \mathbb{Z}$ it means each individual one must be an integer.