+152 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 \(\phi(2^{100}) \)?
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.
+26367 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:
\(\text{If $p$ is prime and $k \ge 1$, then $\\ \phi(p^k) = p^{k-1}(p-1)$ }\)
\(\begin{array}{|rcll|} \hline \mathbf{\phi(p^k)} &=& \mathbf{ p^{k-1}(p-1) } \\\\ \phi(2^{100}) &=& 2^{100-1}(2-1) \\ \mathbf{\phi(2^{100})} &=& \mathbf{2^{99}} \\ \hline \end{array}\)
