Sorry, title should be "multiplicativity", not "multiplicity".
I am supposed to show that, for Euler's totient function:
a.) If n is an odd integer, then .
b.) If n is an even integer, then .
I thought I would use the multiplicity of phi to establish this, i.e.
Obviously this doesn't work, because while it works for (a), it contradicts (b). Not sure why.
Be careful here. The formula only works when This is why it works for (a) but not for (b), as if and only if is odd.
For (b) consider and If is even, the integers in which are coprime with excludes the even integers; hence an integer in is coprime with if and only if it is coprime with For note that is coprime with if and only if is coprime with hence the same argument can be applied to to show that there are integers in coprime with Thus