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What is most motivating way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.

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    $\begingroup$ $\phi (n)$ is also the size of the group of units of $\frac{\mathbb{Z}}{n\mathbb{Z}}$, as a ring. $\endgroup$ – RougeSegwayUser Oct 26 '13 at 14:06
  • $\begingroup$ Equivalently but without any abstract algebra: $\phi(n)$ is the number of integers smaller than $n$ which are coprime relative to $n$. $\endgroup$ – bbnkttp Oct 26 '13 at 14:17
  • $\begingroup$ $\phi(n)$ is also $[\mathbb Q(e^{2\pi i/n}):\mathbb Q]$. $\endgroup$ – Hagen von Eitzen Oct 26 '13 at 14:29
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    $\begingroup$ What do you mean by "real life"? $\endgroup$ – Bruno Joyal Oct 26 '13 at 15:12
  • $\begingroup$ I think the comment given by Chris Dugale is more or less the main point: $\varphi(n)$ is the size of a certain finite group, the units mod $n$, which shows up in many places. If you think the finite matrix groups ${\rm GL}_d({\mathbf Z}/(n))$ are worthwhile then you probably would want to know how big they are. The simplest case $d = 1$ corresponds exactly to the units mod $n$, with size $\varphi(n)$. $\endgroup$ – KCd Oct 26 '13 at 18:58
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RSA, or public-key cryptography is one of them.

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  • $\begingroup$ RSA only needs the case $n = pq$ with distinct primes $p$ and $q$, so it is a very limited use of the $\varphi$-function. $\endgroup$ – KCd Oct 26 '13 at 18:55

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