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

  1. #1
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    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: 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?
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    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.
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    Quote Originally Posted by bumcheekcity View Post
    On another note, is there a sticky anywhere with advice as to the syntax of the [ MATH ] tag?
    Here. The first few posts on the thread have Adobe documents for the most commonly used LaTeX symbols.

    -Dan
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