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