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# can anyone plz help

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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
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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:

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

$$\begin{array}{|rcll|} \hline \mathbf{\phi(p^k)} &=& \mathbf{ p^{k-1}(p-1) } \\\\ \phi(2^{100}) &=& 2^{100-1}(2-1) \\ \mathbf{\phi(2^{100})} &=& \mathbf{2^{99}} \\ \hline \end{array}$$ Jul 9, 2019