Originally Posted by

**particlejohn** So we have $\displaystyle \phi(n) = \prod_{i=1}^{k} \phi(p_{i}^{e_{i}}) = \prod_{i=1}^{k} p_{i}^{e_{i}-1}(p_{i}-1) $. So basically what it is saying that the number of numbers coprime to $\displaystyle n $ is equal to the totient product of their prime factors. So then $\displaystyle \phi(8) = 4 \neq \phi(2) \cdot \phi(2) \cdot \phi(2) $. What's wrong with this (am I interpreting this incorrectly)? Also why is $\displaystyle \phi(n) $ defined for integers $\displaystyle \leq n $ as opposed to just integers $\displaystyle < n $. Because a number is never coprime with itself right?