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Math Help - expansion and factorisation diophantine equation

  1. #1
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    expansion and factorisation diophantine equation

    hey could someone please help me with this question... i think 10 is one of the answers buh i got that through guess and check and not through a proper mathematical method...

    Find all the positive integer solutions to the equation

    X^2 + y^2 + z^2 = 10(x+y+z)

    Prove there are no other solutions


    thankx,,,
    Last edited by phatjigga; February 17th 2007 at 04:28 PM. Reason: not enough information
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  2. #2
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    We have,
    x^2+y^2+z^2=10(x+y+z)
    x^2+y^2+z^2=10x+10y+10z
    x^2-10x+y^2-10y+z^2-10z=0
    (x^2-10x+25)+(y^2-10y+25)+(z^2-10z+25)=75
    (x-5)^2+(y-5)^2+(z-5)^2=75
    a^2+b^2+c^2=75

    By Fermat's 3 square theorem we can express it as a sum of three squares. We do that by trial an error (if you allow non-negatives),
    0+25+36=75

    Thus,
    x=0,y=10,z=11
    x=10,y=0,z=11
    x=10,y=11,z=0
    x=0,y=10,z=10
    x=11,y=0,z=10
    x=11,y=10,z=0
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