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Is there a well defined class of mathematical problems which produce only rational numbers as their solutions?

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Yes, this class of problems is called "division". Specifically, when you divide a whole number by another whole number you get a rational number. That said, I suspect you want something else, so maybe be more specific in your question? –  orlandpm Feb 22 '13 at 18:47
    
@orlandpm An example of such problems would be, if it exists, polynomials having only rational roots. I am aware that some 2nd order polynomials with rational coefficients do so. I am seeking a more general class of problems. –  Tarek Feb 22 '13 at 19:11
    
Such problems are sometimes called Diophantine, although the term is often interpreted more narrowly to apply to integer solutions only, or more broadly, to apply to solutions in some field closely related to the field of rational numbers. –  André Nicolas Feb 22 '13 at 19:26
    
@AndréNicolas In Diophantine problems, you narrow the domain of solutions right from the beginning to integers or rational numbers. What I am asking about is when the only solutions are rationals. –  Tarek Feb 22 '13 at 19:41
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I am not sure one can say much. Linear equations with rational coefficients only have rational solutions. –  André Nicolas Feb 22 '13 at 19:59

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