What is the use of Euler Totient or Phi Function?

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.

• $\phi (n)$ is also the size of the group of units of $\frac{\mathbb{Z}}{n\mathbb{Z}}$, as a ring. – RougeSegwayUser Oct 26 '13 at 14:06
• Equivalently but without any abstract algebra: $\phi(n)$ is the number of integers smaller than $n$ which are coprime relative to $n$. – bbnkttp Oct 26 '13 at 14:17
• $\phi(n)$ is also $[\mathbb Q(e^{2\pi i/n}):\mathbb Q]$. – Hagen von Eitzen Oct 26 '13 at 14:29
• What do you mean by "real life"? – Bruno Joyal Oct 26 '13 at 15:12
• 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 $\varphi(n)$. – KCd Oct 26 '13 at 18:58

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