# prove that φ(n/p^K) is even

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• Mar 26th 2012, 09:42 AM
jojodeco
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
• Mar 26th 2012, 10:01 AM
Sylvia104
Re: Prove that φ(n/p^K) is even
What is the full question? You need to tell us at least what $\displaystyle n$ is. It can't just be any positive integer; for example, the statement is false if $\displaystyle n=p^k.$
• Mar 26th 2012, 10:22 AM
jojodeco
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
• Mar 26th 2012, 10:45 AM
Sylvia104
Re: Prove that φ(n/p^K) is even
Okay. Hint: For any positive integer $\displaystyle m,$ if $\displaystyle \varphi(m)$ is odd, then $\displaystyle m=1$ or $\displaystyle m=2.$