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Math Help - Prove no rational number whose square is 98

  1. #1
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    Prove no rational number whose square is 98

    Q: Prove that there is no rational number whose square is 98

    Here is what i have so far:

    Suppose, for contradiction, that there is a rational number a/b with gcd(a,b)=1 whose square is 98.

    (a/b)^2=98
    a^2=98b^2
    a^2=2(49)b^2

    So, does this show that a is even? how do i show that b is even to get a contradiction?
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  2. #2
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    You are on the right way:

    Suppose, for contradiction, that there is a rational number a/b with gcd(a,b)=1 whose square is 98.

    (a/b)^2=98
    a^2=98b^2
    (a^2 / (49 b^2) = 2
    (a/7b) = sqrt(2) (or (a/7b) = - sqrt(2))
    (a/c) = sqrt(2)
    => sqrt(2) is a rational number
    => Contradiction.
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