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Math Help - x^3+y^3=z^3

  1. #1
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    x^3+y^3=z^3

    Question

    If x^3+y^3=z^3 has a solution in integers x, y, z, show that one of the three must be a multiple of 7.

    How should I solve this question?
    Thank you very much.
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  2. #2
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    Quote Originally Posted by Jenny20 View Post
    Question

    If x^3+y^3=z^3 has a solution in integers x, y, z, show that one of the three must be a multiple of 7.

    How should I solve this question?
    Thank you very much.
    Work modulo 7.
    (7k)^3=7
    (7k+1)^3=7k+1
    (7k+2)^3=7k+1
    (7k+3)^3=7k+6
    (7k+4)^3=7k+6
    (7k+5)^3=7k+6
    (7k+6)^3=7k+6

    Possibilities,
    7k,7k+1,7k+6

    We assume \gcd(x,y,z)=1.
    In that case not all can be multiple of sevens.

    The possibilities that do not lead to contradiction,
    (7k)+(7k+1)=(7k+1)
    (7k)+(7k+6)=(7k+6)
    (7k+1)+(7k+6)=(7k)
    (7k+6)+(7k+1)=(7k)
    We see a multiple of seven does exist.
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