I have this proof - I can do the second direction (starting with "on the other hand") and I understand some of the logic in this first direction...but I can't figure out WHY he did the first step? I want to know mostly why "we can write $n=2^{k}p^{m}$" at the beginning? Also, why does the non-existence of such $p$ imply the result?  Proof:  Firstly, let $\varphi(n)=\frac{1}{2}n$, in which case we can write $n=2^{k}p^{m}$ so that $\varphi(2^{k}p^{m})(1=\frac{1}{2})(1-\frac{1}{p})=\frac{1}{2}2^{k}p^{m}$. But, when attempting to solve for $p$ in the factor $(1-\frac{1}{p})=1$, we find that we want to find $p$ such that$-\frac{1}{p}=0$, for which there exists no number $p$. So, $n$ must contain only $2$s. Thus, $n=2^{k}$ for some $k$.  On the other hand, let $n=2^{k}$ for some integer $k$. Then we can calculate  $\varphi(n)=2^{k}(1-\frac{1}{2})=2^{k}(\frac{1}{2})=\frac{1}{2}n  $
Show that $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k. I think I have it figured and would like to see if I am on the right track. Thank you.
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The answers on this site about this are years old &amp; do not apply to my problem. replacing the default console emulator under Windows XP How can I replace the default program providing console windows under Windows XP?  I know of , and I have a shortcut to start cmd.exe in . But now I want console applications to start in Console2 rather than the default console program, even when I have no control over the program that starts the console application. (I.e. a non-console program starts consoleapp.exe, and I can't change it to start Console2 instead, but I still want the application to be started inside a new instance of Console2.)  (Note that I want to replace the console itself, that is, the window in which console (i.e. text mode) applications run. And I must be able to run arbitrary, unmodified console applications: a substitute for a specific console program such as Cmd won't do me any good.)  EDIT: So what I'm after is a , which           OT: I want to know when Microsoft is going to make a decent CSRSS replacement. Not being able to adjust the width of a "terminal" by resizing the window is a complete joke.         Go download the ISE already. (It's included in Win7/2008R2.)   But as far as I understand this ISE is an , not a general console emulator. Portable Ruby on Rails environment I got myself a new 8 gig USB key and I'm looking for a decent solution to have a portable RoR environment to learn on. I did the google on it and found a few possibilities, but I'm curious to hear some real life experiences and opinions. Thanks! I often hear about the wave-particle duality, and how particles exhibit properties of both particles and waves. However, I wonder, is this actually a duality? At the most fundamental level, we 'know' that everything is made up out of particles, whether those are photons, electrons, or maybe even strings. 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