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I need to prove that

$$\phi(mn) > \phi(m)\phi(n)$$

if $m$ and $n$ have a common factor greater than 1.

I have read up on the case where $m$ and $n$ are relatively prime, then $\phi(mn)=\phi(m)\phi(n)$.

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    Write both $m,n$ as product of power of primes, and observe that $$\varphi(p^k) = p^{k-1}(p-1) > \varphi(p)^k = (p-1)^{k-1}(p-1)$$2012-03-13
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    J.D., this is pretty much the answer OP is looking for. You should post this.2012-03-13
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    It'd be nice to show explicitly that the natural map $U_{mn}\to U_m \times U_n$ is surjective but not injective, if it is true.2012-03-13
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    Ah, it seems that the natural map $U_{mn}\to U_m \times U_n$ has a kernel of size $d$ and an image of index $\phi(d)$ and so my idea above does not seem to work...2012-03-13
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    @Dane This is interesting.2012-03-13
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    @lhf: Your idea works well enough...it leads to Dane's answer. (By the way, I had the same thought of expressing things in terms of this natural homomorphism. I think it would be a positive contribution if you left this as answer.)2012-03-13

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