The Euler 
-function is 
defined by
Here g.c.d. denotes greatest common divisor.
We have not made this function up. It is an important function in number
theory, combinatorics and algebra, and it has sweet properties.
For example, if 
and a,b are coprime (i.e. have
1 as their greatest common divisor), then
In fact this follows immediately from
the next proposition.
Proposition Let the prime divisors of n be
(without repetition). It follows that
Proof The theorem is trivially true if n = 1, since
the empty product is 1. Thus we may assume n > 1, and thus 
For each i in the range 
we put
Here we have eschewed the vertical line notation 
deliberately
in order to avoid notational collision.
Notice that 
Moreover, if 
then
and so on. The inclusion-exclusion
principle enables us to count the natural numbers between 1 and n
(inclusive) which are not coprime to n in two ways.
Rearrange, and a use a little algebra to obtain the result.