Any pattern for when the ring $\mathbb{Z}_m$ has units? ($m$ is any integer)

Question

For example, $\mathbb{Z}_7$ has at least two units, since $2\cdot 4 = 4\cdot 2=1$.

But $\mathbb{Z}_6$ has no units (since nothing multiplies to $7$).

But $\mathbb{Z}_8$ has a unit: $3\cdot 3=1$

So it seems that for $\mathbb{Z}_m$ $($ where $m\in \mathbb{Z})$, if $m+1$ is not prime then $\mathbb{Z}_m$ has units, and if $m+1$ is prime then $\mathbb{Z}_m$ does not have units.

Is this a correct characterization (does it always hold)?

Answer 1

As per the comments, there are always precisely $\varphi(m)$ units, where $\varphi$ is Euler's totient function.

To take your example, $\Bbb Z_6$ has $\varphi(6)=2$ units. They are $1$ and $5$.