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Math Help - iff proof

  1. #1
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    iff proof

    Please Help!

    Prove that n is a square iff the exponent of every prime number occuring in the factorisation of n is even.

    Hence prove that the square root of any natural number is either an integer or an irrational number.

    Any help would be greatly appreicated I don't know anyone else who can.
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  2. #2
    Moo
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    Hello,

    Write the decomposition into prime factors of n.

    n = \prod_{i=1}^k p_i ^{\alpha_i}

    So n^2 = \prod_{i=1}^k p_i ^{\alpha_i} \prod_{i=1}^k p_i^{\alpha_i} = \prod_{i=1}^k [p_i^{\alpha_i} \times p_i^{\alpha_i}]

    As a^b a^c = a^{b+c}, the previous expression equals to :

    \prod_{i=1}^k p_i^{\alpha_i + \alpha_i} = \prod_{i=1}^k p_i^{2 \alpha_i}

    Hence the exponent of every prime number occuring in the factorisation of n is even because multiple of 2.

    And as it's equalities, there is equivalence.
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