Let $m_i > 1$, where $1 ≤ i ≤ n$, be integers, pairwise relatively prime. Let $m = m_1 \cdots m_n$. Let $\phi(m)$ denote the order of the group $(Z/mZ)^×$. The function $\phi : Z_+ → Z_+$ is called the Euler phi function. Show that there exists an isomorphism $(Z/mZ)^× → (Z/m_1Z)^× \times \cdots \times (Z/m_nZ)^×$. In particular, $\phi(m) = \phi(m_1) \cdots \phi(m_n)$.
Attempt: the direction is to use Chinese Remainder Theorem, but I am kinda stuck there at the origin. If mi are all primes, then the result would be immediate, simply by definition. Any hints please?
You must use the Chinese Remainder Theorem. Note for the Chinese Remainder Theorem, exist an surjective homomorphism from $Z→(Z/m_1Z) × · · · × (Z/m_nZ)$, since $m_iZ$ are coprime ideals. The kernel of this homomorphism will be $\bigcap m_iZ=mZ$, then by the first isomorphism theorem you will have a isomorphism beetwen $Z/Z_m$ and $(Z/m_1Z)\times · · ·\times(Z/m_nZ)$, now taking the units of each side of this isomorphism you conclude the result.