Let n be within Z with n>0. Prove that the number of fractions a/b in lowest terms with 0<a/b1 and bn is
The factions are:
$\displaystyle x/1$, [1]
$\displaystyle x/2$, [2]
$\displaystyle x/3$, [3]
...
$\displaystyle x/n$. [4]
In [1] the number of reduced fractions is $\displaystyle \phi (1)$.
In [2] the number of reduced fractions is $\displaystyle \phi (2)$.
In [3] the number of reduced fractions is $\displaystyle \phi(3)$.
...
In [4] the number of reduced fractions is $\displaystyle \phi(n)$.
Thus, in total we have $\displaystyle \phi ( 1) + ... + \phi(n)$ such fractions.