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?
Try more examples and see if you can figure it out. Start by trying to prove it for $\phi(p \cdot p)$, where $p$ is prime. Then, make things a bit more complicated and try again. Eventually, you want to be able to prove it for two general numbers with prime factorization $p_1^{r_1} \cdots p_m^{r_m}$ and $q_1^{s_1} \cdots q_n^{s_n}$. Since the gcd is not 1, you know that they share at least one prime in common. Just let $p_1 = q_1$.