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Thread: 2 Problems

  1. #1
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    2 Problems

    Calculate $\displaystyle \varphi (n)$ for $\displaystyle n = 21^{21} $ and $\displaystyle n=2009 $ In each case, express the answer as the product of prime numbers

    Let a,b be natural numbers. Show that if $\displaystyle a|b $ then $\displaystyle \varphi (a)| \varphi (b) $

    thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Jimmy_W View Post
    Calculate $\displaystyle \varphi (n)$ for $\displaystyle n = 21^{21} $ and $\displaystyle n=2009 $ In each case, express the answer as the product of prime numbers

    Let a,b be natural numbers. Show that if $\displaystyle a|b $ then $\displaystyle \varphi (a)| \varphi (b) $

    thanks
    Here we assume that $\displaystyle \varphi(n)$ denotes Euler's Totient function. In which case to answer this question you need the result that:

    $\displaystyle
    \varphi(n)= (p_1-1)p_1^{k_1-1}(p_2-1)p_2^{k_2-1} .. (p_r-1)p_r^{k_r-1}
    $

    where $\displaystyle n=p_1^{k_1} p_2^{k_2} .. p_r^{k_r}$ is the prime decomposition of $\displaystyle n$.

    CB
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