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Math Help - Can someone help with a proof involving Euler's phi function?

  1. #1
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    Can someone help with a proof involving Euler's phi function?

    Using the formula for \varphi(n), compute \varphi(27),\quad \varphi(81), and \varphi(p^\alpha), where p is a prime number. Give a proof that the formula for \varphi(n) is valid when n=p^\alpha, where p is a prime number.

    Here is what I have so far:

    <br />
27=3^3 ==> \varphi(27)=27\left(1-\frac{1}{3}\right)=18<br />

    <br />
81=3^4 ==> \varphi(81)=81\left(1-\frac{1}{3}\right)=54<br />

    <br />
\varphi(p^\alpha)=p^\alpha\left(1-\frac{1}{p}\right)=p^\alpha-p^{\alpha-1}=p^{\alpha-1}(p-1).<br />

    I am trying to find a way to go about proving what it wants me to prove, but I can not see any methods that will work. Does anyone have any suggestions?
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  2. #2
    Member Traveller's Avatar
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    Hint: \phi (n) refers to the number of positive integers not greater than n, that are relatively prime to n. When n = p^{\alpha} where p is a prime, the only numbers less than or equal to p^{\alpha} that are NOT relatively prime to it are the ones that are divisible by p.
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