In ...

Totient Function -- from Wolfram MathWorld
... is written the following precedure...

... the 'fundamental theorem of arythmetic' extablishes that any integer $\displaystyle m>1$ is represented as the product...

$\displaystyle \displaystyle m= \prod_{k=1}^{r} p_{k}^{\alpha_{k}}$ (1)

... where $\displaystyle p_{k} , k=1,2,...,r$ are the r distincts primes deviding m. If $\displaystyle m= p^{\alpha}$ the numbers of factors relatively prime to m is...

$\displaystyle \displaystyle \varphi (m) = p^{\alpha}\ (1-\frac{1}{p})$ (2)

If $\displaystyle m= p_{1}^{\alpha_{1}}\ p_{2}^{\alpha_{2}}$ the numbers of factors relative prime to m is...

$\displaystyle \displaystyle \varphi (m) = p_{1}^{\alpha_{1}}\ (1-\frac{1}{p_{1}})\ p_{2}^{\alpha_{2}}\ (1-\frac{1}{p_{2}}) = m\ (1-\frac{1}{p_{1}})\ (1-\frac{1}{p_{2}})$ (3)

Proceeding by induction we arrive at the formula...

$\displaystyle \displaystyle \varphi(m)= m\ \prod_{k=1}^{r} (1-\frac{1}{p_{k}})$ (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...

$\displaystyle \varphi(m\ n) = \varphi (m)\ \varphi (n)$ (5)

... is derived from (4) and not viceversa...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$