Euler's Totient Function Proving

lil_luc

I need some help/hints on how to prove this statement. I don't know where to start!


Prove that if m and n are positive integers such that m|n, then φ(mn) = mφ(n).

Thanks

JSuarez

It's a direct consequence of Euler's formula:
[tex]
\phi\left(n\right)=n(1-\frac{1}{p_1})(1-\frac{1}{p_2}) \cdots (1-\frac{1}{p_k})
[/tex]
Because the hypothesis implies that the prime factorization of m has the same factors as n.

Or you can count the numbers that are coprime with nm in [tex]\left\{1,\ldots,mn\right\}[/tex] and prove that there are as many as the ones that are coprime with n in [tex]\left\{1,\ldots,n\right\}[/tex].