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March 26, 2017

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Let a,b,c be positive integers such that a divides b^2 , b divides c^2 and c divides a^2 . Prove that abc divides (a + b + c)7 .

  • urgent -plsee - ,

    Is that a power?

    (a+b+c)^7

    ?

  • urgent -plsee - ,

    Haven't worked through all the details, but I think if you consider that the sum of powers of all the terms in the expansion is 7, just examine every combination.

    Naturally, all the terms with a^x b^y c^z are divisible by abc.

    Consider a^7
    a^2 = mc
    c^2 = nb
    a^7 = a*(a^2)*a^2
    = a*(mc)^2*a^2
    = a*m^2c^2*a^2
    = a*m^2*nb*a^2
    = abc*m^2*n*a^2

    so, it appears that all of the terms will be divisible by abc.

  • urgent -plsee - ,

    Let a,b,c be positive integers such that a divides b^2 , b divides c^2 and c divides a^2 . Prove that abc divides (a + b + c)^7

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