26. Totients.—By the totient of
,
which is denoted, after Euler, by
,
we mean the number of integers prime to
,
and not exceeding
.
If
,
the numbers not exceeding
and not prime to it are
of which the number is
:
hence
.
If
are prime to each other,
;
and hence for the general case, if
,
where the product applies to all the different prime factors of
.
If
,
are the different divisors of
,
.
For example,
.