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Math Help - find (a,b)

  1. #1
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    find (a,b)

    Hello,

    find all (a,b) with a,b \in \mathbb{N}^{*} such that :  \displaystyle \frac {a}{b} + \frac {{21b}}{{25a}} \in\mathbb{N}
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  2. #2
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    Quote Originally Posted by maria18 View Post
    Hello,

    find all (a,b) with a,b \in \mathbb{N}^{*} such that :  \displaystyle \frac {a}{b} + \frac {{21b}}{{25a}} \in\mathbb{N}
    Clearly if (a,b) is a solution then so is (ca,cb) for any natural number c. So the only interesting solutions will be those for which a and b have no common divisor. There are two such solutions.

    Spoiler:
    They are (a,b) = (3,5) and (a,b) = (7,5). Now prove that there are no others.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    Clearly if (a,b) is a solution then so is (ca,cb) for any natural number c. So the only interesting solutions will be those for which a and b have no common divisor. There are two such solutions.

    Spoiler:
    They are (a,b) = (3,5) and (a,b) = (7,5). Now prove that there are no others.
    you are wrong Opalg
    Spoiler:
    (a,b) = (3n,5n),  n \in \mathbb{N}^{*} and (a,b) = (7n,5n),n \in \mathbb{N}^{*}
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  4. #4
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    Maria18, how is your solution different from Opalg's?
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