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

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

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 $\displaystyle a/b, c/d, where a, b, c, d E R$. Then, $\displaystyle p+q = (ad+bc)/bd, p*q = ac/bd$, 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?**