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Math Help - a old german contest problem

  1. #1
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    a old german contest problem

    3.(1977.)How many pairs of numbers p,and q from which are smaller than 100 and for which has a rational solution exist?

    Biggest problem is that I don't know when this type of equation does have a rational solution?

    I do not know how to start?
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  2. #2
    Member grgrsanjay's Avatar
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    there can be infinitely many solutions

    but,I'm not sure
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  3. #3
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    Quote Originally Posted by myro111 View Post
    3.(1977.)How many pairs of numbers p,and q from which are smaller than 100 and for which has a rational solution exist?

    Biggest problem is that I don't know when this type of equation does have a rational solution?

    I do not know how to start?
    Step 1. It follows from the rational root theorem that any rational solution x of this equation must in fact be an integer.

    Step 2. If p and q are natural numbers then x must be negative (if x is positive then the left side of the equation will be positive, not zero). So let y=-x. Then the equation becomes y^5+py=q, and p, q, y are all positive integers, with p and q less than 100.

    Step 3. If y\geqslant3 then y^5\geqslant3^5=243>100, which is larger than q. Therefore y = 1 or 2.

    Step 4. Now all you have to do is to count how many pairs (p,q) there are when y=1 and when y=2.
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