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Math Help - prove that φ(n/p^K) is even

  1. #1
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    prove that φ(n/p^K) is even

    so in this problem i no that gcd(a,n)=1 and the question i need to answer is : let p be any prime factor of n and let k be the number of times that p appears in the prime factorization of n. prove that φ(n/p^K) is even
    so far i have
    If p is prime, then φ(p^k) = p^(k-1)(p-1). So, if p is odd, then p-1 is even, hence
    φ(p^k) is even.
    There is an odd prime q, such that q^k appears in the prime factorization of n.
    Then (q - 1)/φ(q^k)/φ(n). Hence φ(n) is even.
    how can i finish answering the question
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  2. #2
    Member Sylvia104's Avatar
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    Re: Prove that φ(n/p^K) is even

    What is the full question? You need to tell us at least what n is. It can't just be any positive integer; for example, the statement is false if n=p^k.
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  3. #3
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    Re: prove that φ(n/p^K) is even

    well it also say that we should let n be an integer that has at least two distinct odd prime factors and there are no primitive roots of n
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  4. #4
    Member Sylvia104's Avatar
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    Re: Prove that φ(n/p^K) is even

    Okay. Hint: For any positive integer m, if \varphi(m) is odd, then m=1 or m=2.
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