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Math Help - prime-power

  1. #1
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    Post prime-power

    1) Prove that if n is a square, then each exponent in its prime-power decomposition is even.
    and
    2) Prove that if each exponent in the prime-power decomposition of n is even, then n is square.
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  2. #2
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    Quote Originally Posted by Sally_Math View Post
    1) Prove that if n is a square, then each exponent in its prime-power decomposition is even.
    and
    I do the first one and leave the second one for thee to think about.

    (Assuming n>1)

    If n is a square it means n=m^2. Now m>1 and so it can be written as m = p_1^{a_1}...p_k^{a_k} by prime decomposition. This means, n = \left(  p_1^{a_1}...p_k^{a_k} \right)^2 = p_1^{2a_1} ... p_k^{2a_k}. And so exponents in prime decomposition of n are even.

    Similar problem: Once you prove the above problem try proving a stronger result. That n>1 is an m-th power if and only if each prime in the decomposition of n is a multiple of m.
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