In ...
Totient Function -- from Wolfram MathWorld
... is written the following precedure...
... the 'fundamental theorem of arythmetic' extablishes that any integer
is represented as the product...
(1)
... where
are the r distincts primes deviding m. If
the numbers of factors relatively prime to m is...
(2)
If
the numbers of factors relative prime to m is...
(3)
Proceeding by induction we arrive at the formula...
(4)
It is remarkable the fact that the (4) has beeen derived only from (1) [i.e. the fundamental theorem of arithmetic...] without any other hypothesis
This is interesting and I wasn't aware of it, but it isn't completely accurate: only in point (13) arrives Wolfram to the formula, after having used induction and, in (6)-(7)-(8) using a not so straightforward deduction on different primes dividing a given natural number n.
Anyway, is another way to approach this.
Tonio
, so that the formula valid for m and n coprimes...
(5)
... is derived from (4) and not viceversa...
Kind regards