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For a positive integer \(n\)\(\phi(n)\) denotes the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\). What is \(\phi(2^{100}) \)?

 Jul 8, 2019
 #1
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https://en.wikipedia.org/wiki/Euler%27s_totient_function

 Jul 8, 2019
 #2
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OK, young person, here is my best "guess"!

Since n is 2^100, or power of 2, all ODD numbers under 2^100 are "relatively prime" to it, which means: 2^100 / 2 =2^99   numbers that are relatively prime to 2^100.

 Jul 8, 2019
 #3
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thx!!!

 Jul 8, 2019
 #4
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For a positive integer \(n,\ \phi(n)\) denotes the number of positive integers less than or equal to \(n\) that are relatively prime to \(n\).
What is \(\phi(2^{100}) \)?

 

\(\phi(n)\) is Euler's totient function

 

Formula, value for a prime power argument:

\(\text{If $p$ is prime and $k \ge 1$, then $\\ \phi(p^k) = p^{k-1}(p-1)$ }\)

 

\(\begin{array}{|rcll|} \hline \mathbf{\phi(p^k)} &=& \mathbf{ p^{k-1}(p-1) } \\\\ \phi(2^{100}) &=& 2^{100-1}(2-1) \\ \mathbf{\phi(2^{100})} &=& \mathbf{2^{99}} \\ \hline \end{array}\)

 

laugh

 Jul 9, 2019

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