I am supposed to show that, for Euler's totient function:

a.) If n is an odd integer, then $\displaystyle \phi(2n)=\phi(n)$.

and

b.) If n is an even integer, then $\displaystyle \phi(2n)=2\phi(n)$.

I thought I would use the multiplicity of phi to establish this, i.e.

$\displaystyle \phi(2n)=\phi(2)\cdot\phi(n)=1\cdot\phi(n)=\phi(n)$.

Obviously this doesn't work, because while it works for (a), it contradicts (b). Not sure why.