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Let $m_1,m_2\ge1$ be such that $\gcd(m_1,m_2) \ne 1$. Argue that $\phi(m_1*m_2)$ is strictly less that $\phi(m_1)*\phi(m_2)$.

What I have so far:

By example (which I'm not sure if that's how he wants the answer... ):

$$\phi(5*20)=\phi(100)=(2^2-2)*(5^2-5)=40$$

$$\phi(5)=4$$

$$\phi(20)=(2^2-2)*(4)=8$$

$$8*4=32$$

Where $32 \lt 40$. Is this a sufficient way to answer this question? Could someone maybe give me a hint as to how to how I could answer the question in a way other than by example?

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    You have to prove it in general. This only answers it for the specific case.2012-10-08
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    How would I go about doing that?2012-10-08
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    Speaking for myself, if I gave a student an extra credit problem, I would care much more than if it were just homework that the student did it on their own. So, do you have permission from your instructor to ask about this problem on here?2012-10-08
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    I'm not even positive it is extra credit, he just said to do it if we can. Last time he did this he gave us extra credit, but idk if he will again. I tried to do as much of it as I could on my own also. :)2012-10-08
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    More is true: $\phi(mn) = \phi(m) \phi(n) \frac{d}{\phi(d)}$.2012-10-08
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    An example is insufficient, generally speaking. Maybe $m_1 = 5$ and $m_2 = 20$ works and maybe other values work and maybe other values don't work. Maybe the pair you chose is the only pair that works! What you need is a logical proof that shows you that any pair of values satisfying the specified conditions will also satisfy the inequality.2015-02-26

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